The only reason for having a battery is to physically start the car - once running it would run forever and a day without a battery

Charge the battery, start the car and get your volt meter out. You should be getting about 13.8 volts with just the engine running, so thats the first sign, and if you turn the car off and the voltage rises from what you had just seen, its definately the alternator

AND if it is the alternator, dont spend £100 on a new one, get your brush pack replaced and save yourself probably a good £75