Although the Angels did not announce the deal, Texas General Manager Jon Daniels was informed by Hamilton's agent that the career .300 hitter had decided to accept an offer reportedly worth $125 million.
Daniels was quoted on the Rangers' website as saying he had believed Texas had been moving toward being able to sign Hamilton, who became a free agent at the end of the 2012 campaign.
"Apparently not," Daniels said. "It's business and everybody has to make their own calls and he has a family to look out for. Josh has done a lot for the organization and the organization has done a lot for Josh.
"I'm a little disappointed in how it was handled, but he had a decision to make and he made it."
Hamilton was a member of the American League All-Star team in all five of the years he spent with the Rangers. He was named the league's MVP following the 2010 campaign and he was also the MVP of the 2010 AL Championship Series, during which Texas earned its first World Series trip in franchise history.
During the 2010 season he hit a league-leading .359 with 32 home runs and 100 RBI.
Hamilton has been plagued by substance abuse during his career, and the Rangers put a support system in place for him during his days with the club.
After the Rangers captured the ALCS in 2010 they celebrated with soft drinks rather than with champagne in deference to Hamilton's addiction problems.
It is the second straight year the Angels have signed a key member of the Rangers in free agency. Last season they picked up pitcher C.J. Wilson.
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