The 25-year-old has penned a three-year-contract after agreeing terms and passing a medical at Ashton Gate.
Head coach Sean O'Driscoll said: "Frank is somebody I've liked for some time. He's a good height, his distribution is good and he has plenty of experience at Championship level, even with the England full squad.
"We were interested from the outset and then it was just a case of seeing if we could get it over the line.
"We're delighted he's finally signed and he's a great acquisition for the football club."
O'Driscoll added: "I think he's lively and will keep the defence on their toes. He's at an age where the best years are ahead of him.
"He's played a lot of games for someone of his age and most of those games came at the level we're trying to get back to.
"Frank is desperate to play football. It would have been easy for him to stay at Derby County or to move to another Championship club, but we've got a number one spot available and he's taken this opportunity."