Without doubt the alternator at fault, nothing else could cause this.
I bet if you hooked up a dig gauge it would show this as you drove.
The warning light is a INDICATION only, it is is possible for some faults to leave a charging alternator even thought he light is on, or not charge even when the light is off, depends what part has gone, in your case the regulator rectifier, but cheap to do a recon should be about £45, as nothing special, the bist on the last one I did were £28, so for the hassle now I just let them do the work and get a year warranty.
Got to a commercial type place for best prices.
tabetha