DoW Lowen said:
<spoiler=Again>Well, I guess it's more of a yes and no.
You do kill him, via an assassination quick-kill. However, at the time both Ezio and the Pope (Borgia) possess a Piece of Eden. The Pope is immediately resurrected via his Piece of Eden, the Papal Staff, at which point stage one of the final boss fight begins.
Stage two of the fight leads you downstairs where Ezio and Borgia have one last fight, this time unarmed. At the end of the fistfight, Ezio tells Borgia he will not kill him, and Borgia is left alive.
So yes, you kill him the first time, but he resurrects, and no, you allow him to live the second time.