Doak leaves Liverpool for a fee of £20m plus £5m in bonuses. He has signed a deal to 2030, with Liverpool also holding a buy-back clause.
The Scotland attacker, 19, told the club's website: "I'm buzzing to be here.
"I feel like it's the right time for me and it's a great opportunity to play regular football at the highest level. It feels like the perfect fit.
"The team play very nice football. It's fast and it really suits my game, so I felt like I couldn't say no to the chance to play here."
Tiago Pinto, the club’s Head of Football Operations, shared his delight with afcb.co.uk: “Ben is an exciting young talent who has shown his quality at the highest level despite his age.
“He brings pace and energy that will fit our model well and we’re thoroughly looking forward to helping him fulfil his potential at AFC Bournemouth.
“We’re pleased to bring another player into the club during this window as we continue to reinvest in the squad.”