The team made the announcement during Wednesday night's game against the Boston Red Sox.
The 32-year-old Cozart went for a second opinion on his shoulder after experiencing continued discomfort. He plans to have surgery in Los Angeles soon.
A free agent last offseason, Cozart left the Cincinnati Reds to sign a $38 million, three-year contract with the Angels and agreed to switch positions from shortstop to third base. He played in just 58 games during his first season with Los Angeles, batting .219 with five homers and 18 RBIs.
"I think this has been battling him on and off for a little bit, I mean in previous years. And it got to the point where something needs to be done, and for Zack it's the best course of action, for our organization the best course of action, and he'll be 100 percent for next year," manager Mike Scioscia said after his team's 9-6 loss to the Red Sox.
Cozart spent his first seven years in the big leagues with Cincinnati.