Yup, had this with my 2006 MacBook, and a new battery fixed it.
(Incidentally, when I last went to buy a replacement battery from an Apple Store, they wouldn't let me install it myself even though the polycarbonate MacBooks had user replaceable batteries. They insisted I bring in my MacBook and leave it with them overnight to have a Genius install it. I ended up cancelling it and getting a NewerTech battery replacement, which is both cheaper & has better battery life than the Apple model.)
(Incidentally, when I last went to buy a replacement battery from an Apple Store, they wouldn't let me install it myself even though the polycarbonate MacBooks had user replaceable batteries. They insisted I bring in my MacBook and leave it with them overnight to have a Genius install it. I ended up cancelling it and getting a NewerTech battery replacement, which is both cheaper & has better battery life than the Apple model.)