From "The Council of Elrond":
Quote:
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"I was at the Battle of Dagorlad before the Black Gate of Mordor, wehere we had the mastery: for the spear of Gil-galad and the Sword fo Elendil, Aiglos and Narsil, none could withstand. I beheld the last combat on the slopes of Orodruin, where Gil-galad died, and Elendil fell, and Narsil broke beneath him; but Sauron himself was overthrown, and Isildur cut the Ring from his hand with the hilt-shard of his father's sword, and took it for his own." -Elrond
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Sauron killed Elendil and Gil-galad, but at the same time they overthrew Sauron - "killing" him, per se. So Sauron was already overthrown by the time Isildur cut off ring and finger.