Cheaper if you're a bit handy with a really small screwdriver and sensible enough to take it easy and earth yourself, to buy a new battery and have a go yourself.
IF it's still the original case / motherboard combination, then
capacity on the back = 2nd Gen
No capacity on back =1st Gen.
Batteries are interchangeable, i.e. not gen specific, but 2nd gen are designed for the batteries to last much longer and nothing you can do about that.
You can however buy more powerful batteries than the originals - just make sure you get a kosher one, and you can swap the microdrive CF type II for a CF type I card. No mechanical parts = less lektrickery used......
p.s. - there are potential problems with the motherboard that may cause power problems and if that's the case you'll need a new motherboard...however, if it's still on its original battery then it probably just needs a new battery - good luck!!
1st Gen Mini - turbo'd to 850mah battery and 32GB CF Type 1
Lovin' my iPod experience
...and now it's GOLD
....AND..ZombiePod - 20GB 4G photo, bought back from the dead (...mostly)
.....AND....dead 1st gen 4GB Nano - soon to be resurrected