The 23-year-old will move to the Aviva Premiership club in time for next season following 17 appearances for Bedford, whom he signed for last year from Blackheath.
"I'm really looking forward to joining the club and I can't wait to meet all the boys come pre-season," Stephenson told the club's official website. "Richard Hill is a great coach so hopefully I will learn a lot from him and continue my progression."
Hill added: "We are trying to build for the future and James is a real talent who is hungry and the right age. This year in the RFU Championship he has shown what an excellent player he is and Paul Hull has watched him closely.
"James is a player with huge potential to get even better and he has shown his undoubted try scoring ability at Bedford Blues. He is fast and he's very powerful, and being an ex-back row forward he likes the contact so is an aggressive and strong winger."
Stephenson is the eighth player signed by the club for next season.