Rome adopted a similar ranking of the ancient sees (Rome 1st, Constantinople 2nd, Alexandria 3rd, Antioch 4th and Jerusalem 5th) at the Fourth Lateran Council in 1215 and the Council of Florence in 1439, but from what I understand, Rome never officially accepted the Canon 3 that was written at the First Council of Constantinople in 381.
From the Fourth Lateran Council:
5. The dignity of the patriarchal sees
Renewing the ancient privileges of the patriarchal sees, we decree, with the approval of this sacred universal synod, that after the Roman church, which through the Lord’s disposition has a primacy of ordinary power over all other churches inasmuch as it is the mother and mistress of all Christ’s faithful, the church of Constantinople shall have the first place, the church of Alexandria the second place, the church of Antioch the third place, and the church of Jerusalem the fourth place, each maintaining its own rank. Thus after their pontiffs have received from the Roman pontiff the pallium, which is the sign of the fullness of the pontifical office, and have taken an oath of fidelity and obedience to him they may lawfully confer the pallium on their own suffragans, receiving from them for themselves canonical profession and for the Roman church the promise of obedience. They may have a standard of the Lord’s cross carried before them anywhere except in the city of Rome or wherever there is present the supreme pontiff or his legate wearing the insignia of the apostolic dignity. In all the provinces subject to their jurisdiction let appeal be made to them, when it is necessary, except for appeals made to the apostolic see, to which all must humbly defer.
From the Council of Florence:
We also define that the holy apostolic see and the Roman pontiff holds the primacy over the whole world and the Roman pontiff is the successor of blessed Peter prince of the apostles, and that he 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 the full power of tending, ruling and governing the whole church, as is contained also in the acts of ecumenical councils and in the sacred canons.
Also, renewing the order of the other patriarchs which has been handed down in the canons, the patriarch of Constantinople should be second after the most holy Roman pontiff, third should be the patriarch of Alexandria, fourth the patriarch of Antioch, and fifth the patriarch of Jerusalem, without prejudice to all their privileges and rights.