Galea, 34, had previously announced his retirement having failed to agree terms with Rovers following five years at Craven Park.
But he will now hook up at the KC Stadium with Peter Gentle, who Galea previously worked with at Wests Tigers.
Galea said: “I am really looking forward to getting started with Hull FC, it is a move that excites me.
“I know all about the club from their rivalry with Hull KR, I’ve been involved in the big derby games and there is nothing quite like that atmosphere at the KC Stadium.
“The longer I have thought about it, the more I’ve realised how much I want to continue with my footy and that I feel I’ve still got enough left in the tank to go around again, so when the chance came up to move to a club like Hull and work with a coach like Peter Gentle, it reinvigorated me and I jumped at it.”
Hull coach Gentle added: “I think Ben is a real coup for us, he’s been such a good player through the years and a champion person that will bring us a wealth of experience to our side, both on and off the field.
“I am in no doubt he has at least another season in him and for us to get him for the next 12 months will have a great effect on our side, particularly the younger players who will learn so much from him.”