There’s one problem with running your MacBook with the battery removed. MacBooks and MacBook Pros automatically reduce the processor speed by about half when the battery is removed.
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 $129 from The Apple Store http://store.apple.com/us/product/MA566LL/A?mco=MTY3ODQ5OTY
Thank you very much for the help!
Today I tested with the battery and it works at full speed!
Now I need to buy a new battery.
I already bought one of those third party batteries for my other Macbook 13'' 2.1 and I'm very happy with it (been using it for more than one year, already made 167 cycles with it).
Anyway I don't want to spend 126$ on a old computer...