My Audi shut off and won't start

I was driving down the road, and it just shut off.
Luckily, it was near a parking lot. When I try to start
it it doesn't do anything. The battery is brand new,
and isn't dead. Head lights, radio, ect all work.
When I turn the key it sounds like the starter tries
to do something for less then a second then
literally no response.