Adams signed a five-year deal with Bournemouth, the club confirmed on Saturday, after a one-year stint at Leeds that ended with him having hamstring surgery in March and the team being relegated from the Premier League at the end of the 2022-23 season.
ESPN reported earlier this week that the two clubs had agreed a fee for in the region of £20 million ($25.4m) for the USMNT midfielder.
Adams is Bournemouth's eighth signing in this transfer window under new manager Andoni Iraola.
"We're thrilled to have brought Tyler to the club and he is a player we've admired for a long time," Bournemouth chief executive Neill Blake said in a statement.
"It's been widely documented how much his talents have been recognised in this transfer window by other clubs, so to have him sign for AFC Bournemouth shows the ambition we have as a club.
"Tyler is an exciting player who possesses plenty of experience for someone that is just 24 years of age and we're looking forward to working with him as he's a player we know will be a great addition to our squad."