iphone 5 won't retain any charge.

Has anyone else experienced this and how do I fix it?

It's only a week old but I don't live near any genious bar so bringing it in isn't an option. I've tried charging it from my computer and wall but even though it turns on and says its fully charged as soon as I unplug it it turns off and the symbol of no battery shows up. All updates are installed so I really don't know what else to do..