My phone just did that today as well. I haven't even had my iphone5 for 18 hours!
I chatted with support and someone from support gave me a call.
If doing a "Restore" (holding town the on/off switch, then holding down the home -- let go of the on/off button after 10 seconds and plug into computer/itunes while still holding the home button -- doesn't do anything, then you'll need to drive to an Apple Store and get a replacement phone. Or, you can talk to support and they'll send you a replacement phone, put a hold on your credit card and there could possibly be a fee invoved as well, even though it's a defective phone, and they'll send you out a new phone in 3-5 business days. With the Holiday's being here, it'll most likely be 5-7 days.
Bummer, isn't it? The closest Apple Store to me is 45 minutes away and NOT at all handy.
-- Unhappy iPhone 5 owner --