I'm not finding any replacement batteries for sale, but I did find [http://www.ebay.com/itm/Eclipse-180G2-PL-Purple-4GB-Digital-Media-Player-MicroSD-Parts-Repair-Plays-/152707259760?hash=item238e0fdd70:g:wuMAAOSwknJXxfDC|one device on ebay] they're selling for someone to use as parts. Looks like the battery still works, along with most other components. You might try just transferring the motherboard/battery combo from that one.
As far as actually replacing the battery, do be aware that it's wired directly onto the motherboard. Someone in the comments says it's just a button cell battery, but if you look closely at pictures of it, it sure doesn't look like one. Here's the guide: [guide|36971]
Good luck!