Battery or Alternator??

Car sat for about a week with the cell phone charging and on return the vehicle is having charging problems i.e. voltage is dropping.  I cleaned the negative post that was pretty corroded.
 
The battery was low.  The voltage when I attached to charger was near zero but got it to fully charge. 
 
When starting, the voltmeter stays stable but the battery indicator flashes.
 
Tried to determine if battery was cause or alternator by disconnecting the negative on the battery.  Old School  so may have been wrong method.  Car ran for around a minute and then cut off.  Tried  a couple of times and same result. 
 
Based on this, I am assuming that it is the alternator.
 
I guess, pure guess, that it could still be the battery if there is a fuel or other mechnism (sp??) that needs the battery to operate.  Maybe electric fuel pump.
 
Any suggestions.  I don't want to take the alternator off if it is the battery.  Really can't afford to take to the shop for diagnosis so any help would be of great help and would be highly appreciated.
 
 

Problem solved.  Alternator was bad. 
 
Interesting however.  I had it checked at autozone including the battery.  Both tested OK.  Took alternator off and had rechecked and it failed all tests.
 
Testing on the car isn't the best idea I guess.

HOW MUCH DID THE ALT COST?