My tech knowledge is not as good as some LOC members, but seeing as no-one else has chipped in - here's my thoughts.
Given the battery is fairly new, and that you don't have any issues starting the car and you've got 12v at the battery terminal, then the most likely explanation is that there is a problem with the charging system.
The alternator may not be bad, more likely the belt is loose or worn, that needs to be checked before spending big cash.
There's some useful info here to get you started
http://www.askthemechanic.co.uk/electrical/charging-system