A good alternator should charge around 14V, I would say an absolute minimum should be 13,5V. Normaly they are labeled 14,8V.
If this is the case, I would check; cable from alternator to battery, AND engine grounding back to the battery.
I had problems with the alternator and regulator so I trew them both out and bought a newer alternator with an internal regulator(easier to wire up) it should be easy to find one that fits.
If the battery holds its charge over night I would call it good, you can test it yourself, if you sort it with a spanner or something while you measure the voltage and the voltage doesn't drop below 10-11V it is fine, don't take to long with this if you decide to do it.
If the car dies while you are driving and you stil meassure 12,3volts on the battery I would think the problem is elsewhere...