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Your Mac is very old, so you should consider the option of buying a new PC or Mac. Anyway, first try to reset the SMC > http://support.apple.com/kb/HT3964
If it doesn't work, the battery has stopped working, so if you don't want to buy a new computer, just get a replacement battery from the Apple Online Store > http://store.apple.com/us/product/MA561LL/A/rechargeable-battery-13-inch-macbook -white
You can get a new third party battery fairly cheap on eBay. Every MacBook from 2006 through Mid 2009 except the Late 2008 Aluminum Unibody and the Late 2009 and 2010 White Unibodies use the same A1185 battery.
Or for $129 from The Apple Store http://store.apple.com/us/product/MA566LL/A?mco=MTY3ODQ5OTY