Before going to the trouble and cost of replacing parts:
1. Verify that you have all updates installed, particularly Battery Update 1.4 (KB# HT3421)
2. Reset the SMC (KB# HT3964)
3. Reset your PRAM (KB# HT1379)
SMC Reset in a Nutshell
1. Remove the battery and AC power
2. Press and hold the power button for 10 seconds
3. Reinstall battery and power
Given this model's track record its probably the battery but SMU's do go on the fritz every now and again (plus resetting the SMU doesn't cost any green).
All MacBook batteries are the same so don't worry about getting your model number (ie. if you really wanted you could put a black battery in a white MacBook, dalmatian style).
Selected battery issues are covered by a Repair Extension Program for which no model is eligible after 3 years from the date of purchase. Eligibility must be determined by an Apple Authorized Technician and repairs fulfilled at option by Apple.