Try using a good battery charger rather than beating your alternator to
death.

The cheapest fix may be charging with a low amp charger and then try
using the battery. It might work fine for you.

Then, it may not hold a charge or have very little capacity when charged.

It would take much, much longer running the car to charge it than using
a charging unit. Alternators are meant to replenish the capacity, not
charge to 100% from zero charge while the battery is in use.

If you ran it down without recharging to 100% over and over and then
left it to discharge completely, it may be sulfated beyond repair.

If you have a charger with a desulfication function you may have some
luck. But it takes time.

Batteries in cold climates can freeze if in a very low state of charge,
and if it has frozen, the case may be cracked someplace. Check all cells
for liquid.

Lastly of course, you may have to purchase a new battery.

The only way to tell if the battery is toast is to recharge it and
see.  You can jump the car to start it and depend on the cars
electrical system to recharge the battery but that might not be the
best way to handle the recharge.

Today's modern car batteries are not intended to be discharged
completely.  They give the best service life when they are used for
short busts of power, like starting the car, and then immediately
recharged.

A battery that is completely discharged and left to sit for an
expended period of time will most likely be junk.  They can sometimes
be saved if handled properly.  The plates will sulfate and you have to
recombine that sulfate back into solution.  You can do this,
sometimes, by an extended slow charge.  Use a trickle charger for a
month or more and then test the battery.

Jack

It depends on whether it's maintenance-free or low-maintenance, but I
don't know why.  The latter are more common and always have removable
caps, and I've had good luck restoring them with a 10A charger run for
5-8 hrs.  OTOH a maintenance-free battery (may or may no have
removable caps) will usually die after just a single deep discharge,
as I learned, again and again.

Recharging a completely dead battery with just the alternator may
overload it, so I would rather use a charger, at least for the first
few hours.  I once left the headlights on overnight and jump started
the dead battery.  The alternator output measured almost 14V after the
jumper cables were unhooked and even 30 minuteslater,  but next
morning the alternator wouldn't read higher than 12.5V (not a cold
morning).  But the battery lasted another 1-2 years.