Rover Mini Cooper sport, 1999, 80 000 miles

My Rover mini is having some problems with charging. We have replaced the battery but it's still giving us trouble. The car will start fine if the battery is fully charged but after a few runs it will only bump start. When the car is idling the voltmeter shows 13 volts and the car appears to be running fine. When driving, the volt meter can dip down as low as 10 and then back up to 13 as soon as the car is idling again. The alternator belt seems tight enough and the lights do not appear dim. The earth for the alternator also appears to be sound. Any suggestions would be most welcome.