Hi,
Sounds like it could be the battery but it's unusual for a Zune to just shut off like that. First thing to try is to leave it plugged into your PC trying to charge for at least 12 hours. This will sometimes kick start the battery back to life.
If you got the Zune from us (ZuneThings) then we honour the warranty. Just send an email and we can get it replaced.
If you got it somewhere else contact them to try and get them to help you. They might if you make enough noise!
Otherwise there is a good repair site in the US that we used in the past and they also have DIY repair guides on the site. Details here:
http://news.zunethings.co.uk/2009/12/zune-repair-uk.htmlCheers,
Rob