His nine-year stay at Turf Moor ended with promotion to the Premier League, and he opted not to re-sign as his contract expired.
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Instead the 33-year-old striker will remain in the second tier with David Wagner's Norwich, and he's intent on gaining promotion again.
"I am delighted, over the moon that it's finally done," he said.
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"I think the ambition to get back into the Premier League was the main thing.
"The conversations I had (with Stuart Webber and David Wagner) were brilliant, they showed me how they wanted me to play and how they want me to fit in to the group and I'm happy to totally push us forward and try to get us promoted next season."
Wagner added: "We're delighted to welcome Ashley to the club. He is a player I've come across on a number of occasions and he knows exactly what it takes to be successful in this league.
"It was very clear from our conversations that he has a real hunger and desire to continue to perform at the highest level. It was a feeling that we both have, the determination to do whatever it takes to win football games.
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"There was a lot of competition for his signature, which says a lot about how he is respected and valued in the game.
"I'm very happy to have this deal completed early and we look forward to welcoming Ashley to the group when we return for pre-season."
During Burnley's promotion campaign under Vincent Kompany, Barnes managed 39 appearances, scoring six goals and registering three assists.