The 27-year-old was expected to leave the newly-demoted Championship side following reports that the shot-stopper had handed in a transfer request last week.
"I'm delighted to be here to start a new challenge and I'm looking forward to it," Bunn told the club's official website.
"I'm looking forward to seeing some familiar faces and getting to know the other lads. It's a good time for Norwich. It will be a tough season and I think we'll do well."
Canaries boss Chris Hughton added: "I'm delighted to have signed Mark, he is a keeper that will give us valuable experience and competition at first team level.
"We are fortunate to have two very good young goalkeepers and it will hopefully give us an opportunity to give them experience on loan."
The contract includes an option for a further year.