Initial X-rays did not show much damage, but a follow-up CT scan revealed the fracture. Sweeney will be placed on the disabled list and the Cubs will call up a replacement before Tuesday's game at Oakland.
"It's unfortunate," Cubs manager Dale Sveum said. "We're losing a big part of our offense. It's unfortunate for him, too. He's trying to get his career going, getting a chance to play every day, and it stinks when those things happen to guys like that."
Sweeney was hitting .295 with four homers and 15 RBIs since his call-up from Triple-A Iowa on May 6. Brian Bogusevic replaced him in center field for Sunday's series finale against the Mariners.
Sveum said he's fine with Bogusevic in that spot for the time being and isn't sure if the team will call up another position player on Tuesday. That largely depends on how many innings starting right-hander Edwin Jackson lasts on Sunday.
Alfonso Soriano was back in left field on Sunday after hitting as the designated hitter the past two games.