~Note: The bolded teachings below would not be true if any Pope were permitted by the grace of God to teach heresy, to err gravely on doctrine in any way, to err gravely on discipline, or to fail in faith by apostasy, heresy, schism, or idolatry. If a Pope or Council could teach heresy, the teachings below would not be true. But they are the infallible teachings of an Ecumenical Council. So these teachings must be true.
Pius, bishop, servant of the servants of God, with the approval of the Sacred Council, for an everlasting record.
1. The eternal shepherd and guardian of our souls , in order to render permanent the saving work of redemption, determined to build a Church in which, as in the house of the living God, all the faithful should be linked by the bond of one faith and charity.
2. Therefore, before he was glorified, he besought his Father, not for the apostles only, but also for those who were to believe in him through their word, that they all might be one as the Son himself and the Father are one .
3. So then, just as he sent apostles, whom he chose out of the world , even as he had been sent by the Father , in like manner it was his will that in his Church there should be shepherds and teachers until the end of time.
4. In order, then, that the episcopal office should be one and undivided and that, by the union of the clergy, the whole multitude of believers should be held together in the unity of faith and communion, he set blessed Peter over the rest of the apostles and instituted in him the permanent principle of both unities and their visible foundation.
5. Upon the strength of this foundation was to be built the eternal temple, and the Church whose topmost part reaches heaven was to rise upon the firmness of this foundation .
6. And since the gates of hell trying, if they can, to overthrow the Church, make their assault with a hatred that increases day by day against its divinely laid foundation, we judge it necessary, with the approbation of the Sacred Council, and for the protection, defense and growth of the Catholic flock, to propound the doctrine concerning the 1. institution, 2. permanence and 3. nature of the sacred and apostolic primacy, upon which the strength and coherence of the whole Church depends.
7. This doctrine is to be believed and held by all the faithful in accordance with the ancient and unchanging faith of the whole Church.
8. Furthermore, we shall proscribe and condemn the contrary errors which are so harmful to the Lord’s flock.
On the institution of the apostolic primacy in blessed Peter
1. We teach and declare that, according to the gospel evidence, a primacy of jurisdiction over the whole Church of God was immediately and directly promised to the blessed apostle Peter and conferred on him by Christ the lord.
2. It was to Simon alone, to whom he had already said You shall be called Cephas , that the Lord, after his confession, You are the Christ, the son of the living God, spoke these words:
Blessed are you, Simon Bar-Jona. For flesh and blood has not revealed this to you, but my Father who is in heaven. And I tell you, you are Peter, and on this rock I will build my Church, and the gates of the underworld shall not prevail against it. I will give you the keys of the kingdom of heaven, and whatever you bind on earth shall be bound in heaven, and whatever you loose on earth shall be loosed in heaven  .
3. And it was to Peter alone that Jesus, after his resurrection, confided the jurisdiction of Supreme Pastor and ruler of his whole fold, saying:
Feed my lambs, feed my sheep .
4. To this absolutely manifest teaching of the Sacred Scriptures, as it has always been understood by the Catholic Church, are clearly opposed the distorted opinions of those who misrepresent the form of government which Christ the lord established in his Church and deny that Peter, in preference to the rest of the apostles, taken singly or collectively, was endowed by Christ with a true and proper primacy of jurisdiction.
5. The same may be said of those who assert that this primacy was not conferred immediately and directly on blessed Peter himself, but rather on the Church, and that it was through the Church that it was transmitted to him in his capacity as her minister.
6. Therefore, if anyone says that blessed Peter the apostle was not appointed by Christ the lord as prince of all the apostles and visible head of the whole Church militant; or that it was a primacy of honor only and not one of true and proper jurisdiction that he directly and immediately received from our lord Jesus Christ himself: let him be anathema.
On the permanence of the primacy of blessed Peter in the Roman pontiffs
1. That which our lord Jesus Christ, the prince of shepherds and great shepherd of the sheep, established in the blessed apostle Peter, for the continual salvation and permanent benefit of the Church, must of necessity remain for ever, by Christ’s authority, in the Church which, founded as it is upon a rock, will stand firm until the end of time .
2. For no one can be in doubt, indeed it was known in every age that the holy and most blessed Peter, prince and head of the apostles, the pillar of faith and the foundation of the Catholic Church, received the keys of the kingdom from our lord Jesus Christ, the savior and redeemer of the human race, and that to this day and for ever he lives and presides and exercises judgment in his successors the bishops of the Holy Roman See, which he founded and consecrated with his blood .
3. Therefore whoever succeeds to the chair of Peter obtains by the institution of Christ himself, the primacy of Peter over the whole Church. So what the truth has ordained stands firm, and blessed Peter perseveres in the rock-like strength he was granted, and does not abandon that guidance of the Church which he once received .
4. For this reason it has always been necessary for every Church–that is to say the faithful throughout the world–to be in agreement with the Roman Church because of its more effective leadership. In consequence of being joined, as members to head, with that see, from which the rights of sacred communion flow to all, they will grow together into the structure of a single body .
5. Therefore, if anyone says that it is not by the institution of Christ the lord himself (that is to say, by divine law) that blessed Peter should have perpetual successors in the primacy over the whole Church; or that the Roman Pontiff is not the successor of blessed Peter in this primacy: let him be anathema.
On the power and character of the primacy of the Roman Pontiff
1. And so, supported by the clear witness of Holy Scripture, and adhering to the manifest and explicit decrees both of our predecessors the Roman Pontiffs and of general councils, we promulgate anew the definition of the ecumenical Council of Florence , which must be believed by all faithful Christians, namely that the Apostolic See and the Roman Pontiff hold a world-wide primacy, and that the Roman Pontiff is the successor of blessed Peter, the prince of the apostles, true vicar of Christ, head of the whole Church and father and teacher of all Christian people.
To him, in blessed Peter, full power has been given by our lord Jesus Christ to tend, rule and govern the universal Church.
All this is to be found in the acts of the ecumenical councils and the sacred canons.
2. Therefore we teach and declare that, by divine ordinance, the Roman Church possesses a pre-eminence of ordinary power over every other Church, and that this jurisdictional power of the Roman Pontiff is both episcopal and immediate. Both clergy and faithful, of whatever rite and dignity, both singly and collectively, are bound to submit to this power by the duty of hierarchical subordination and true obedience, and this not only in matters concerning faith and morals, but also in those which regard the discipline and government of the Church throughout the world.
~ Note: if this required submission were only concerning infallible teachings, then it would not also regard discipline and government. Therefore, the submission includes the non-infallible in both doctrine and discipline.
3. In this way, by unity with the Roman Pontiff in communion and in profession of the same faith , the Church of Christ becomes one flock under one Supreme Shepherd .
4. This is the teaching of the Catholic truth, and no one can depart from it without endangering his faith and salvation.
5. This power of the Supreme Pontiff by no means detracts from that ordinary and immediate power of episcopal jurisdiction, by which bishops, who have succeeded to the place of the apostles by appointment of the Holy Spirit, tend and govern individually the particular flocks which have been assigned to them. On the contrary, this power of theirs is asserted, supported and defended by the Supreme and Universal Pastor; for St. Gregory the Great says: “My honor is the honor of the whole Church. My honor is the steadfast strength of my brethren. Then do I receive true honor, when it is denied to none of those to whom honor is due.” 
6. Furthermore, it follows from that supreme power which the Roman Pontiff has in governing the whole Church, that he has the right, in the performance of this office of his, to communicate freely with the pastors and flocks of the entire Church, so that they may be taught and guided by him in the way of salvation.
7. And therefore we condemn and reject the opinions of those who hold that this communication of the Supreme Head with pastors and flocks may be lawfully obstructed; or that it should be dependent on the civil power, which leads them to maintain that what is determined by the Apostolic See or by its authority concerning the government of the Church, has no force or effect unless it is confirmed by the agreement of the civil authority.
~Note: Some would have it that decisions of the Supreme Pontiff have no force unless it is confirmed by the agreement of the conservative subculture, or by individual Catholic’s comparison between the decision and “the tradition”.
8. Since the Roman Pontiff, by the divine right of the apostolic primacy, governs the whole Church, we likewise teach and declare that he is the supreme judge of the faithful , and that in all cases which fall under ecclesiastical jurisdiction recourse may be had to his judgment . The sentence of the Apostolic See (than which there is no higher authority) is not subject to revision by anyone, nor may anyone lawfully pass judgment thereupon . And so they stray from the genuine path of truth who maintain that it is lawful to appeal from the judgments of the Roman pontiffs to an ecumenical council as if this were an authority superior to the Roman Pontiff.
~Yet so many Catholic media outlets, priests, theologians, and bloggers put their own personal judgment forward as if it were “an authority superior to the Roman Pontiff”.
9. So, then, if anyone says that the Roman Pontiff has merely an office of supervision and guidance, and not the full and supreme power of jurisdiction over the whole Church, and this not only in matters of faith and morals, but also in those which concern the discipline and government of the Church dispersed throughout the whole world; or that he has only the principal part, but not the absolute fullness, of this supreme power; or that this power of his is not ordinary and immediate both over all and each of the Churches and over all and each of the pastors and faithful: let him be anathema.
On the infallible teaching authority of the Roman Pontiff
1. That apostolic primacy which the Roman Pontiff possesses as successor of Peter, the prince of the apostles, includes also the supreme power of teaching. This Holy See has always maintained this, the constant custom of the Church demonstrates it, and the ecumenical councils, particularly those in which East and West met in the union of faith and charity, have declared it.
2. So the fathers of the fourth Council of Constantinople, following the footsteps of their predecessors, published this solemn profession of faith: The first condition of salvation is to maintain the rule of the true faith. And since that saying of our lord Jesus Christ, You are Peter, and upon this rock I will build my Church , cannot fail of its effect, the words spoken are confirmed by their consequences. For in the Apostolic See the Catholic religion has always been preserved unblemished, and sacred doctrine been held in honor. Since it is our earnest desire to be in no way separated from this faith and doctrine, we hope that we may deserve to remain in that one communion which the Apostolic See preaches, for in it is the whole and true strength of the Christian religion .
~The above teaching excludes grave errors by the Roman Pontiff.
What is more, with the approval of the second Council of Lyons, the Greeks made the following profession:
“The Holy Roman Church possesses the supreme and full primacy and principality over the whole Catholic Church. She truly and humbly acknowledges that she received this from the Lord himself in blessed Peter, the prince and chief of the apostles, whose successor the Roman Pontiff is, together with the fullness of power. And since before all others she has the duty of defending the truth of the faith, so if any questions arise concerning the faith, it is by her judgment that they must be settled.” 
Then there is the definition of the Council of Florence:
“The Roman Pontiff is the true vicar of Christ, the head of the whole Church and the father and teacher of all Christians; and to him was committed in blessed Peter, by our lord Jesus Christ, the full power of tending, ruling and governing the whole Church.” 
~If you do not treat every Pope as the true Vicar of Christ and as the father and teacher of all Christians, including yourself, you are the one who has failed gravely in faith, not the Pope.
3. To satisfy this pastoral office, our predecessors strove unwearyingly that the saving teaching of Christ should be spread among all the peoples of the world; and with equal care they made sure that it should be kept pure and uncontaminated wherever it was received.
4. It was for this reason that the bishops of the whole world, sometimes individually, sometimes gathered in synods, according to the long established custom of the Churches and the pattern of ancient usage referred to this Apostolic See those dangers especially which arose in matters concerning the faith. This was to ensure that any damage suffered by the faith should be repaired in that place above all where the faith can know no failing .
5. The Roman pontiffs, too, as the circumstances of the time or the state of affairs suggested, sometimes by summoning ecumenical councils or consulting the opinion of the Churches scattered throughout the world, sometimes by special synods, sometimes by taking advantage of other useful means afforded by divine providence, defined as doctrines to be held those things which, by God’s help, they knew to be in keeping with Sacred Scripture and the apostolic traditions.
6. For the Holy Spirit was promised to the successors of Peter not so that they might, by his revelation, make known some new doctrine, but that, by his assistance, they might religiously guard and faithfully expound the revelation or deposit of faith transmitted by the apostles.
Indeed, their apostolic teaching was embraced by all the venerable fathers and reverenced and followed by all the holy orthodox doctors, for they knew very well that this See of St. Peter always remains unblemished by any error, in accordance with the divine promise of our Lord and Savior to the prince of his disciples: I have prayed for you that your faith may not fail; and when you have turned again, strengthen your brethren .
7. This gift of truth and never-failing faith was therefore divinely conferred on Peter and his successors in this See so that they might discharge their exalted office for the salvation of all, and so that the whole flock of Christ might be kept away by them from the poisonous food of error and be nourished with the sustenance of heavenly doctrine. Thus the tendency to schism is removed and the whole Church is preserved in unity, and, resting on its foundation, can stand firm against the gates of hell.
~The whole flock can only be kept away from the poisonous food of error, i.e. grave errors, if this gift of truth and never failing faith to Peter protects him from teaching and committing heresy at all times, not only when he exercises infallibility.
8. But since in this very age when the salutary effectiveness of the apostolic office is most especially needed, not a few are to be found who disparage its authority, we judge it absolutely necessary to affirm solemnly the prerogative which the only-begotten Son of God was pleased to attach to the supreme pastoral office.
9. Therefore, faithfully adhering to the tradition received from the beginning of the Christian faith, to the glory of God our savior, for the exaltation of the Catholic religion and for the salvation of the Christian people, with the approval of the Sacred Council, we teach and define as a divinely revealed dogma that when the Roman Pontiff speaks EX CATHEDRA, that is, when, in the exercise of his office as shepherd and teacher of all Christians, in virtue of his supreme apostolic authority, he defines a doctrine concerning faith or morals to be held by the whole Church, he possesses, by the divine assistance promised to him in blessed Peter, that infallibility which the divine Redeemer willed his Church to enjoy in defining doctrine concerning faith or morals. Therefore, such definitions of the Roman Pontiff are of themselves, and not by the consent of the Church, irreformable.
~The above bolded teachings do not refer solely to papal infallibility. The Charism of Truth and of Never Failing Faith is a perennial teaching of the Magisterium.
So then, should anyone, which God forbid, have the temerity to reject this definition of ours: let him be anathema.
Given at Rome in public session, solemnly held in the Vatican Basilica in the year of our Lord one thousand eight hundred and seventy, on the eighteenth day of July, in the twenty-fifth year of Our Pontificate.
In conformity with the original.
Joseph, Bishop of St. Polten Secretary to the Vatican Council
1 The Profession of faith of the other fathers added: and I pledge and swear true obedience to the Roman Pontiff, successor of blessed Peter the Prince of the Apostles, and Vicar of Jesus Christ
2 The profession of faith of the other fathers continues: my subjects, or those for whom I have responsibility in virtue of my office, hold, teach and preach the same
3 See Mt 28, 20.
4 See Heb 13, 9.
5 1 Tm 2, 4.
6 Lk 19, 10.
7 Jn 11, 52.
8 See Wis 16, 12.
9 Is 59, 21.
10 See Lateran Council IV, const. 1 (see above, p. 230).
11 Wis 8, 1.
12 Heb 4, 13.
13 Rm 1, 20.
14 Heb 1, 1-2.
15 1 Cor 2, 9.
16 Council of Trent, session 4, first decree (see above p. 663).
17 Heb 11, 1.
18 Mk 16, 20.
19 2 Pt 1, 19.
20 Council of Orange II (529), canon 7 (Bruns 2, 178; Msi 8, 713).
21 Heb 11, 6.
22 Is 11, 12.
23 1 Tm 2, 4.
24 1 Pt 2, 9; Col 1, 13.
25 Col 1, 12.
26 Heb 2, 3.
27 Heb 12, 2.
28 Heb 10, 12.
29 Rm 1, 20.
30 Jn 1, 17.
31 i Cor 2, 7-8, 10.
32 Mt 11, 25.
33 2 Cor 5, 6-7.
34 See Lateran Council V, session 8 (see above p. 605).
35 See Col 2, 8.
36 Vincent of Lerins, Commonitorium (Notebook), 28 (PL 50, 668).
37 1 Pt 2,25.
38 Jn 17, 20-21.
39 Jn 15, 19.
40 Jn 20, 21.
41 Leo 1, Serm. (Sermons), 4 (elsewhere 3), ch. 2 for the day of his birth (PL 54, 150).
42 Jn 1, 42.
43 Mt 16, 16 19.
44 Jn 21, 15-17.
45 See Mt 7, 25; Lk 6, 48.
46 From the speech of Philip, the Roman legate, at the 3rd session of the Council of Ephesus (D no. 112).
47 Leo I, Serm. (Sermons), 3 (elsewhere 2), ch. 3 (PL 54, 146).
48 Irenaeus, Adv. haeres. (Against Heresies) 1113 (PG 7, 849), Council of Aquilea (381), to be found among: Ambrose, Epistolae (Letters), 11 (PL 16, 946).
49 Council of Florence, session 6 (see above p. 528).
50 See Jn 10, 16.
51 Ep. ad Eulog. Alexandrin. (Letter to Eulogius of Alexandria), VIII 29 (30) (MGH, Ep. 2, 31 28-30, PL 77, 933).
52 Pius VI, Letter Super soliditate dated 28 Nov. 1786.
53 From Michael Palaeologus’s profession of faith which was read out at the second Council of Lyons (D no. 466).
54 Nicholas I, Ep. ad Michaelem imp. (Letter to the emperor Michael) (PL 119, 954).
55 Mt 16, 18.
56 From Pope Hormisdas’s formula of the year 517 (D no. 171), see above p. 157 n. 1.
57 From Michael Palaeologus’s profession of faith which was read out at the second Council of Lyons (D no. 466).
58 Council of Florence, session 6 (see above p. 528). S Bernard, Ep. (Letters) 190 (PL 182, 1053).
59 Bernard, Ep. (Letters) 190 (PL 182, 1053).
60 Lk 22, 32.
The translation found here is that which appears in Decrees of the Ecumenical Councils ed. Norman Tanner. S.J. The numbering of the canons is however found in Tanner’s text.