The 28-year-old, who joined Stoke from Portsmouth in 2010, has signed for a fee reported in British media to be worth 8 million pounds.
Begovic will compete with first-choice Belgium goalkeeper Thibaut Courtois at Stamford Bridge after Petr Cech ended his trophy-laden 11-year spell by joining rivals Arsenal last month.
"I am very happy to be joining Chelsea. After speaking at length to the manager (Jose Mourinho), I feel like I can develop here and be an important part of this team," Begovic told Chelsea's official website (www.chelseafc.com).
Begovic becomes Chelsea's second signing of the close season after Colombian striker Radamel Falcao joined from AS Monaco on a season-long loan deal.
(Reporting By Michael Hann; editing by Amlan Chakraborty)
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