put a volt guage on the battery when it is running and see if you're putting out  14-15 volts running. It should. If so, alternator should be fine. 
Cold weather is a battery's worse nightmare. If the battery is a few years old or more, it may be getting weak.  Trying to start an engine in the cold is putting quite a shock to a battery that has seen its better days.  Try turning on the headlights for a minute or so, then turn them off.  Then crank the engine.  That should get the juices flowing before putting such a shock to the battery.  At least, that is the theory.