The midfielder was available for no fee following his departure from QPR, where he spent last season in the Championship.
A red card in May's play-off final rules the 31-year-old out of Norwich's opening game against Wolves this weekend.
He told the City website: "I'm really pleased. I trained with the lads this morning and I'm just ready to get going now.
"It's a good club that has still got loads of good players, so hopefully we'll have a good year. They're keen to get back to the Premier League, they have fantastic support and everyone I've spoken to has said what a great club it is and what a great area it is to live.
"I'll get to Wolves to watch the lads and it's important that we hit the ground running."
Norwich boss Neil Adams said: "Gary’s a very experienced player. He’s featured in the Premier League and Championship all his career and he can play in a number of roles through the midfield.
"He was with QPR when they achieved promotion to the Premier League, so he's precisely the sort of player needed. The other players have been really impressed with how he's applied himself in the three or four training sessions he's had with us so far."