The 24-year-old has penned a five-year contract with the Cherries, keeping him at the club until 2028.
Adams joined the Whites last summer from RB Leipzig in a reported £20 million deal.
He hit the ground running at Elland Road, starting 24 of their first 27 Premier League matches.
But then he picked up a serious hamstring injury in March that ruled him out for the remainder of the campaign.
The midfielder still hasn’t made full recovery but his undoubted potential sparked an interest from Bournemouth who have now got a deal over the line.
The United States international previously looked to be heading to Chelsea after they agreed to trigger the £20m release clause in his contract at Elland Road.
But the Blues pulled out of the move as they wanted to pursue deals for Moises Caicedo and Romeo Lavia, with the Londoners now having successfully completed transfers for both players.
Adams mainly plays as defensive midfielder but can also be deployed as a right midfielder or a right-back.
His departure is a big blow for the West Yorkshire outfit who have already lost several of their key players this summer.
Daniel Farke’s side have been far from convincing in the Championship so far. Two draws and a 1-0 defeat at Birmingham means they have picked up only two points from their opening three matches.
The Whites now face a tricky trip to Ipswich Town, with the match available to watch via Leeds live streams.
The Tractor Boys have made an impressive start to life in the Championship after earning promotion to the second-tier.
They have won all their three games and are currently at the top of the standings.
Farke’s men must come up with something special to be able to take full points from the Portman Road.
- ICC Cricket World Cup 2023: Full squad lists of all 10 teams
- Preview: Tottenham Hotspur vs Liverpool – prediction, team news, line-ups
- Memsie Stakes – A Look at the Past Winners
- Five League Cup matches to watch this week
- Preview: Chelsea vs Aston Villa – prediction, team news, line-ups
- Preview: Arsenal vs Tottenham Hotspur – prediction, team news, line-ups
- Aston Villa eye Europa Conference League success after impressive 2022-23 season
- Could Pep Guardiola’s Manchester City Go Unbeaten This Season?
- Chelsea starlet agrees new contract ahead of potential January exit
- West Ham eye another European trophy after Conference League success last season