I'd be surprised if the battery has failed already -- mine, a March 2005 car which is used pretty well every day, and locked and unlocked several times a day, is still using the original battery.
But, the answer is that the lower part of the fob pops open - with the key blade 'open' you can pull the case apart by pulling quite hard at the join under the Fiat logo. Here's what the manual shows:
Pete