The 26-year-old, who can play wing or full-back, will join the Championship outfit at the end of the season.
Amesbury told Sky Sports News: "I'm excited about joining Bristol - a club that is clearly building for the future. Liam Middleton impressed me with the style of rugby and type of culture that he is building there.
"There is lots that I want to achieve in my career and I'm confident I'm coming to a place where I can develop my game and help Bristol achieve Aviva Premiership status."
Amesbury has played 13 times for Sale this season, scoring two tries.