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 Uni 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