The speculation linking the Netherlands international with a move to the Etihad Stadium had gathered pace in recent days, with the defender arriving in Manchester to undergo a medical.
City have now officially announced the 25-year-old on a deal until the summer of 2025.
"City have been the best side in England over the course of the last decade. Coming here is a dream for me. This is a top side full of world-class players. Everywhere you look in this squad there are big names with international pedigree," Ake told the club's official website.
"Pep [Guardiola] is a manager admired across the world – what he's done in the game speaks for itself. The success he's had is unbelievable and the style of football he plays really appeals to me.
"I know I'm going to have to work hard to get into the side, but that's what I'm here to do. I'll do whatever I can to make an impact and help the team win silverware."
Ake made 30 appearances for Bournemouth during the 2019-20 campaign - including 29 in the Premier League - but was unable to prevent the Cherries from dropping into the Championship.
The Dutchman, who has allegedly cost Guardiola's side £41m, becomes City's second signing of the summer after Spanish attacker Ferran Torres arrived from Valencia.