A couple of things dont seem right to me.... either its a weird battery failure or like has been mentioned something is draining the battery fast. But for a car to just stall with all the lights on? Hmmm maybe the battery failed and caused the voltage to go to high without the load and shutdown. But then there is load caused by the ignition system. Did the abs/airbag lights stay on when the clock went off? Would have to be a major fire starting fault to kill a battery that quick.
Try a new battery as a first point of call. There's nothing to loose
Also check how hot the battery is cause from memory shorted cells from build up of deposits cause quite a bit of heat. And just because the car runs one minute doesnt mean the battery is ok as the charging from the alternator might be whats triggering the battey off and you are just running on the power provided by the alternator.