The Bosnia-Hercegovina international joins Chelsea on a four-year deal.
Stoke had hoped to keep the 28-year-old, who had one year remaining on his contract at the Britannia Stadium.
“I have loved my time at Stoke but feel I had to take this opportunity to continue to develop and challenge myself as a player,” Begovic said.
Begovic, who joined the Potters from Portsmouth in 2010, is the champions’ second signing of the summer following the arrival of Radamel Falcao on a season-long loan.
“I am very happy to be joining Chelsea,” said Begovic.
“After speaking at length to the manager, I feel like I can develop here and be an important part of this team. I am looking forward to meeting up with the team on Wednesday for the pre-season tour.”
Begovic, who will challenge Thibaut Courtois for the main goalkeeper role, will take the number one shirt left vacant following the sale of Petr Cech to Arsenal.
 
source: bbc