"This was always going to be a difficult decision," he said.
"But a chance to join one of the world's top broadcasters on day one of a new channel was too good to reject."
Head of BBC TV Sport Philip Bernie said: "Ray has been a fantastic member of the BBC Sport team for many years and we wish him all the best for the future."
ESPN acquired the live rights for 46 Premier League matches next season following the collapse of Irish broadcaster Setanta.
Stubbs's first match fronting their coverage will be Everton v Arsenal at Goodison Park on 15 August.
