Over the last few weeks I could tell my car was turning over sluggishly
when starting......like when battery is dying....  Then 3 times the last
few days, after cranking my car, my radio had lost power...preset
channels gone, clock not set, but the interior clock in my car had not
reset.  Today after sluggishly cranking the car and driving it, I cut it
off upon returning home and car would not start.  Lights, horn, interior
lights all worked fine, starter would not engage....just the fast rat
tat tat of when turning over.  I connected jumper cables to my car and
car started fine, but would not restart without jumper cables attached.  
Therefore, I assumed a bad/dead battery.....not enough juice to turn
over motor.

I replaced the battery, the car starts just fine now.  However, now my
battery light comes on when idling and from 800-1800 rpm's.  The engine
now revs up and down by itself from 800-1800 rpm's and battery light
dims and brightens with rev'ing of the engine.  Once I increase rpm's
above 1800 rpm's, the battery light goes off and all seems normal.

What's my problem.....???   I suspect the alternator is in process of dying.

HELP!!!

Thanks.