The 20-year-old has been linked with a number of clubs during the summer after submitting a transfer request to the Bairns, with Wigan Athletic and Aberdeen both keen on the Scottish Under-21 star.
But Miller visited Ashton Gate earlier this week to discuss terms and has now penned a three-year deal at the club, with the option of a fourth year if both parties agree.
The Robins will have to pay in excess of £300,000 for the highly-rated frontman but feel it is money well spent for a player with international pedigree.
Scottish Division One side Falkirk had hoped that the youngster would stay for another season, but Miller made it clear that he wanted to move to an English club.
And, after meeting with manager Danny Wilson and being shown round the stadium earlier in the week, he has opted to ply his trade with the West Country club.
