Rivers, a 6-foot-2, 235-pound linebacker who spent four seasons with the Bengals, was acquired in exchange for an undisclosed pick in the 2012 NFL Draft.
A former first-round pick out of USC, Rivers missed the entire 2011 season while rehabilitating from wrist surgery, but registered 77 tackles and one sack in 2010.
The 25-year-old said he is happy for an opportunity to wipe the slate clean.
"I'm excited to get a fresh start and brand new beginning," Rivers said. "You can't ask for more than to be playing for the defending world champions and be in New York."
"This is an excellent opportunity for him to have a fresh start," coach Tom Coughlin said. "Keith is versatile enough to play the outside on first and second down and he can play on third down because he runs well. And he's fast enough to rush the passer."
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