The Cherries revealed on Twitter on Friday they had snapped up the defender:
According to the club's official website, he has signed a "long-term deal" at the Vitality Stadium.
The Guardian's Stuart James reported the fee is thought to be £20 million, while the contract is for five years.
Ake spent time on loan at Bournemouth last season before being recalled by Chelsea in January. Per the club's official website, he said:
"I had a great time here last season so I'm really happy to be back.
"It's an exciting challenge. Chelsea is a great club and I learned so much there, but this is a new step and the right place for me to develop even more.
"The fans haven't seen the best of me yet. I always set the bar high so I know there is a lot of things I still have to work on.
"With the help of the manager here I know I can keep improving and become a better player."
The 22-year-old also took to Instagram to thank his former club:
The left-footed centre-back enjoyed a positive spell with the Cherries last year. Squawka Football shared the numbers behind his impressive but brief stint:
Despite his youth, Ake already looks to be a reliable defender who anticipates danger well and works hard to regain possession for his side.
While he made just five appearances for Chelsea upon his return to Stamford Bridge in January, he earned his first senior caps for the Netherlands at the end of the season and will likely go on to win many more if he fulfils his potential.
His reported fee is likely to raise some eyebrows, but he's a promising talent and could prove to be an excellent capture for Bournemouth if he continues his development with them in the coming years.