Brett, yes, the alternator light should go out immediately.  However I would
check for a loose or worn belt first.  If the belt is worn, then loosen and
remove the belt from the pulley of the alternator.  Now try to move the
pulley from side to side (9 oclock to 3 oclock) and up and down (12 oclock to
6 oclock).  If the pulley moves or you hear a clicking noise, then it is time
for a rebuilt alternator.  This trick also works on waterpumps and while the
belt is loose this is also a good time to check the waterpump (it will seem
fine if the belt is tight but will flop like crazy when the belt is off if it
is bad).
I have rebuilt Bosch Alternators for $150 with a $75 core (do it once, do it
right!).  New Waterpumps are $50 and I include clamps, hoses, and seals with
the pumps.

Hope this helps,
Ken Wilford
http://www.vanagain.com
John 3:16
Office (856)-765-1583
Shop (856)-327-0027
Fax   (856)-327-2242