well you could always
and this is not the best way to test this
but...
while the car is running if you pulled off the ground cabel to the battery.... if the car keeps runnig it might be your loose connection... but if it sputteres and dies, it is your alt... but that would be the sure fire way to know
It does sound like a bad alternator, mine did that too when it finally died. could drive it about 5 minutes and it would kill (this was also because the battery was bad too) the symtoms you are saying would lead to indicate an alt.
lastly just check the 1 plug into the alt, see if there is alot of black on the terminals, and you could check to make sure it is grounded the same way (pull off the 10mm with the wires run to it) and see if they are blackend or corroded
but I would assume an alternator (and a new battery in the future maybe
)
ps as a side note... your battery sounds like it is also getting bad because you should be able to run your car for a decent amount of time with just the battery powering everything but it can't charge without the alt, so it will eventually die