Chelsea signed Begovic from Stoke City in 2015 after allowing Petr Cech to join Arsenal.
However, the Bosnia international has struggled to make his mark at Stamford Bridge, making just 17 Premier League starts.
Begovic has spent the majority of his time at Chelsea behind Thibaut Courtois in the pecking order.
But he will now vie with Artur Boruc for the No1 spot at Bournemouth next season, with the Cherries signing him for an undisclosed fee.
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10. Fraser Forster – £10m (Celtic to Southampton)
A Chelsea statement said: "Asmir Begovic will join Bournemouth in a permanent transfer on 1 July.
"The 29-year-old's professional attitude, friendly nature and ability to perform when called upon made him a popular figure with Chelsea fans, team-mates and staff.
"We thank Asmir for his contribution over the past two years and wish him well in his future career."
Begovic said: "AFC Bournemouth is an exciting, ambitious club which has everything in place for a very bright future.
"Being part of a title-winning team at Chelsea was amazing but I felt now was the right time to move on because I want to be playing regular football.
"I am coming into an important part of my career and I want to show people what I am capable of.
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Chelsea goalkeeper Asmir Begovic will be at Bournemouth next season
"When I spoke to Eddie Howe I knew AFC Bournemouth was the right club for me and I want to bring my experience to the squad, both on and off the pitch.
"I’m really excited to be part of the journey here and hope I can help take the club to another level."
Cherries boss Eddie Howe added: "Asmir is an outstanding goalkeeper who has proved himself capable of performing at the highest level over a number of years.
"He is at the right age and the right fit. He is a leader as well and exactly the kind of character you want in the team.
"Asmir will provide great competition to the excellent group of goalkeepers we already have here and his experience will prove invaluable in the dressing room and on the pitch.
"I look forward to him being a very important player here for years to come."