Q: After alternator is replaced, battery light still comes on

Last week while driving, for about 3 seconds the battery light flickered on and then off. The voltmeter jumped up and down between high and low. Two days ago, the battery light appeared again and this time it stayed on. The meter kept low and then started dropping lower and lower. I was thinking that the problem was probably the alternator and as I was trying to get to my house the instruments began to go out. First the tach, then the speedometer, and then the accelerator pedal was unresponsive. I was able to limp for a little while, but it eventually totally quit. After jump-starting the truck, I made it to the mechanic’s shop. They tested the battery and it was fine until the engine was on and then it was only getting 12 volts, not the 13 or 14 that it should have been getting. They reported it was a bad alternator and then replaced it. The truck performed well throughout the next day until the battery light came on again. This time the voltmeter was at the proper place, about 2/3rd way up and approximately 13 or 14 bolts. Later in the evening, the light came on again but only below 1500 RPM and the meter was ⅓ and around 1 volt. I returned to the auto repair shop again and discovered that they had given me a rebuilt alternator - a bad rebuilt one. This time they gave me a brand new alternator. While going home, the light faintly appeared again! The voltmeter is now staying around 12 volts. All of this makes me think that the alternator is not the problem. Can you tell me what “is” the problem and also tell me how to fix it? Thank you