probally this is going to be the first time someone hear about this:

last month i rebuild my alternator because the light was comming on
with the engine at low speed, it cost me 100$Canadian to do it, first
week the car was perfect, but sutently the battery light start one more
time to come one, but this time not only at low speed!!!!. (my
alternator mecanic told me one day that the way that honda civic's are
made, with all the acc. on and with the negative cable of the battery
off the car should shut off(only with acc. on: lights, heather,
etc.)... soo i stop the car on a parkin lot and with the acc. OFF i
pull off the negative cable of the battery, the car stay! on!!!!, my
ALT. was working. with out worring i drove back to my house. next day i
put my multimetter with the engine off on the battery it say 11.98V it
was correct, i start the car, the battery light stay on, and i check
again  the V. it was 14.30V... what the F.1@! the light is on and the
alternator is working!. i read the repair book and it say to check some
of the fuse like: battery fuse, under dash #12 and #24( the #12 i never
founded). but i check all the fuses and they were OK!. soo. i decide to
do somme experiences: i ran 2 cables from the battery to inside the car
by the window and i connceted them with the multimetter. the result
were:
with the car off and acc's on: 11.10V + - .30V
car off and light off: 12.10V
car on light off (car not moving): 14.30V
car on light off (highter that 10km/h): 14.35V
car on all acc.on (car not moving): 12.80V + - 1.00V
car on all acc.on (car moving more that 10km/h): 14.30V ALWAYS!

soo that is happening with that light!!!!! that light is driving me
crazy!, i check the connection on the ALT. i even clean them and the
are perfect!. i check the hayness diagrams but i cant find what is the
cable that controls that light. because probaly there is a bad
connection somewere. IF THERE IS SOMEONE THAT HAD A SIMILAR PROBLEM OR
THAT HAS A MORE PRECISE ELECTRIC DIAGRAM BESIDE THE ONE OF THE HAYNESS
REPAIR MANUAL THAT WE GET AT CANADIAN TIRE FOR 20$( BECAUSE THAT ON IS
NOT! PRECISE). LET ME KNOW. and for the people that will probaly reply
by saying that is a altenator of a battery problem is not!. we are in
december and there in quebec we start the day with temp of -13C and let
me tell you, every single morning my car start at the first try, even
with 250 000 km on it. i have a other car, a tercel of the same year
they are fantastic this 2 japans cars they had NEVER let me down even
with temps under -25C. they don't even burn oil. soo any info regarding
the F.1@! light let me know.

Here is an update.

Today I had the alternator replaced at a mechanic.  $125 labor + $75
for the part.
The first thing I did when I got the car is grabbed my multimeter

Car off: 12.2 V
Car idle: 14.5 V

So I thought.  Great! It's all fixed.  (The battery light was not on
... everything perfect)

I drove a few kilometers and the red light reappeared.  I popped the
hood to do another measurement.

Car off: 12.2 V
Car idle: 12.5 V

So after a few km I have the same problem again?   The one thing that
did improve versus the state prior to the change is that now at least
when I increase the trottle I do get more juice.  As much as 13V ...
but that's still a bit low.  Before no matter how many RPMs I was
running the engine at it was always steady at 11.2V

Now I also notice a rubber burning / electric motor burning smell when
I pop the hood.  What does that mean?  Bad alternator?  Belt too tight
somehow?
The engine also sounds different.  It has a sort of a whizzing sound
that increases with RPMs .. could that be the new alternator?  Is that
ok?

Looks like I'm not out of the woods yet.  Did the mechanic do a shitty
job?  Did they put a crappy alternator in there?

He told me I might have to replace the battery?  Could that be it?
Could it be that the previous problem degraded my battery to the point
that it won't charge anymore?  (I did drive my car to the mechanic this
morning successfully - with the battery light on)