Is the battery for the iPad 2 compatible with the iPad 1?

Hello,

I have a 64GB 3G/GPS iPad 1st Gen (A1337) of which boot-loops whenever I start charging it and try to turn it on. I am currently trying to find a replacement battery for it and cannot find one that looks legitimate. I looked on the store here at iFixIt but the oldest gen iPad battery replacement I could find was a replacement for the iPad 2. So I am wondering if I buy the iPad 2 replacement battery if it will work with my iPad 1 and if not where can I buy a replacement.

Thank you,

-Jenson