The Grizzlies announced the deal with the 37-year-old Carter early Saturday morning, but did not disclose terms.
Sources told ESPN.com that Carter was finalizing a three-year, $12.2 million contract Friday night to join the Grizzlies, who have coveted him for some time.
The Mavericks were hoping to keep Carter but didn't have the financial flexibility to pay him the sort of money Memphis offered after extending a three-year, $46 million offer sheet to Houston Rockets restricted free agent Chandler Parsons.