You didn't by chance accidentally switch your power adapter thingie with one from a MacBook, did you?
'Dirty Girls'
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If it's powering your computer when attached, I'd start to suspect the battery or the charger.
The battery appears good, I don't know how to check the charger.
I've had this happen with three batteries in MacBooks. Basically if you leave it plugged in most of the time you aren't allowing the battery to fully cycle and it will eventually fail and stop charging. The good news is that in all of the cases where I've had this happen a trip to the Apple Genius Bar has netted me a new battery at no charge.
I'm almost certain this is your charger, quester. I had almost the exact same effect happening with my old Macbook Pro charger - really faint green light that kept the computer running, but didn't charge the battery. Another charger would work perfectly.
I was usually able to get mine working by unplugging and replugging it, turning it over and trying the other direction, and moving the wire around a bit - generally futzing with it. Eventually, the light would turn charging orange and all would be well with the world. But it was getting worse. Then I spilled Diet Dr. Pepper on that computer and bought a new one, so I haven't had to worry about it since.
So Gris, by charger, do you mean the the power cord with the mag plug? Because replacing that would be easier than a battery, I'm thinking.
Either way take it to an Apple store. In almost every instance that I've had to deal with something like this they've given me a new battery or new power cord gratis. This is even on machines that are long out of Apple Care coverage.
Seconding ND - the Genius Bar employees are empowered to give you free shit if they're in a good mood, and they almost always are.
I would be always in a good mood if I worked at a Genius Bar, too...
Quester, the charger is actually inside the base of the MacBook, generally it is a built-in AC adapter, and shunts power to both charging and powering the laptop.