EJP, I think "the words he wrote" were the chorus of the song: "Fare thee well, for I must leave thee, etc." However, that brings up an interesting point that both parties are ostensibly singing the chorus and, while it might make sense for him to sing "Fare thee well for I must leave thee/ Do not let this parting grieve thee/ And remember that the best of friends must part, must part" but the "I'll hang my heart on the weeping willow tree" would probably be her words. I've always felt that her singing "remember that the best of friends must part," was kind of a reluctantly bitter comment. She'd prefer that they not have to part. It's not necessarily just her saying goodbye to her friends because she's about to die. Think of "Look down look down that lonesome road/ Hang down your head and cry/ The best of friends must part some day/ So why not you and I," which also seems to be a song about one lover not being faithful to another. Maybe the beginning part of the chorus is his words that she's bitterly parroting back at him. So instead it'd read:
Him: "Fare thee well for I must leave thee/ Do not let this parting grieve thee/ Remember that the best of friends must part."
Her: "Adieu, adieu kind friends adieu, yes adieu/ I can no longer stay with you, stay with you/ I'll hang my heart on the weeping willow tree/ And may the world go well with thee."
But obviously all that's just speculation on my part.