The Rays selected Driscoll's contract from Triple-A Durham on Sunday, Ryan Bass of Bally Sports Sun reports.
The 26-year-old has an .840 OPS with seven homers in 70 games with Durham this season, and he'll now join the Rays for his first taste of the big leagues. Driscoll isn't likely to see regular playing time and should operate as the No. 3 catcher while Ben Rortvedt and Alex Jackson are healthy.