To my way of thinking, I would replace the suspect battery first and see how the alternator responds to that..a short drive will tell the story.
Depending on the brand alternator and it's individual capabilities, it is possible for it to go into overdrive and cook itself, trying to keep up.
Just to help clarify what I mean....my Nissan Patrol came equipped with two HD 12v batteries and an alternator to match. After a heavy day in the mud, winching every couple of hundred yards or so, these batteries would be way down on capacity...my fitted voltmeter would sit around the 18v mark for more than an hour on the drive home, then gradually creep back down to about 14v.
It became a topic of conversation during a club meet one day and we ended up comparing alternators fitted to similar model vehicles. From the outside, they all looked the same...but a check of the little specs plate attached to each one revealed a difference.
It may well be that your alternator is toasted because of trying to keep up with a dead, or dying battery, but I don't really think that the alternator would cause the battery to die. I would change the battery first...even if it's a known good one from a mates car...a short drive will do no harm and may just reveal the truth.
Worth a try, just to save you some money.