Weems spent his first five seasons with Atlanta and was the NFC's special-teams selection for the Pro Bowl in 2010.
Weems has 41 career special-teams tackles, 77 punt returns for 815 yards and 113 kickoff returns for 2,896 yards. He also has 24 receptions for 205 yards.
McCown got his deal one day after Chicago agreed to a one-year contract with Jason Campbell to back up Cutler. The Bears also acquired Pro Bowl receiver Brandon Marshall in a trade with Miami on Tuesday, signed special-teams standout and linebacker Blake Costanzo from San Francisco and agreed to a two-year contract with cornerback Tim Jennings that kept him off the free-agent market.