If your alt light is on when the engine's not running that's normal (in fact, if it doesn't come on when the engine's not running, the alternator won't work!). If it comes on at idle, that could indicate a problem. If it comes on at higher revs that's definitely a problem.
I say it might be a problem at idle if you're running a non-stock alternator. An alternator for a different car may need to be turned faster to generate power, so when placed in a mini it may not work at idle. If you still have the stock alternator, I don't think this is the case.
Get another battery and jump start the car. If it still won't start, it's not the power supply that's the problem. If it starts up and runs fine, check:
- All wiring to the alternator
- Fanbelt tension
- Battery terminals are clean and tight
If you're still having problems, you'll really need a multimeter to check it out. You can then check the charging voltage and the alternator resistance (the alt resistance should be close to zero).
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