Murphy captained the Tigers in the Guinness Premiership and Heineken Cup finals as the curtain came down on Corry's career.
Leicester head coach Richard Cockerill said: "Geordan has been at Tigers for more than a decade and has been a fantastic servant to this club.
"He was captain at the end of last year and did a fantastic job.
"The way we want to play, and the way we want to improve our game, is with ball in hand and Geordan epitomises that.
"He has the respect of the rest of the squad and I wanted someone who is integral to Leicester Tigers, which he is."
Murphy captained Leicester for the first time in the league win over Sale Sharks in April and guided the Tigers to victory in the Premiership grand final.
"It is a great honour to lead out such a great club but there is so much experience and so much quality around the playing squad that it allows you to get on with your own game and trust the rest of the guys to do the same," he said.
Meanwhile, Leicester have completed four signings, including the much-travelled Irish fly-half Jeremy Staunton from Wasps as cover for the injured Toby Flood.
Staunton is joined at Welford Road by hooker Joe Duffey and prop Robbie Harris from Nottingham while centre Billy Twelvetrees has signed from Bedford.
Cockerill said: "Jeremy Staunton has been capped by Ireland and we are looking for that goal-kicking with Toby Flood being injured.
"He gives us another experienced option in the backline positions and a proven ability as a goal-kicker."
Staunton has previously played Premiership rugby for Harlequins, London Irish and Wasps.
The 28-year-old scored 23 points in Wasps' Premiership victory at Welford Road last season, a record for any opposition player in a league match at the ground.