and hope it dont fry your ecu
werd..... do not disconect your battery while the engine is running. You might be able to get away with it for about 30 seconds before your car shuts down. This is a trick the old carburated guys use, its not a good idea for your efi car. interuptng the power to your ECU is never a good idea. Ive seen it not literally fry an ECU but, erase it . Its almost like pulling the plug on your PC while you have 3 or more programs running.
When it comes to your problem, I cannot stress how important it is to own a volt meter. If you dont have one, $12 will pick you up one from any auto parts store. Charge your battery DISCONNECTED FROM THE CAR COMPLETELY and have the tester on hand. Check the battery to make sure it has about 12.6 volts. If it doesnt, go buy a battery. If it does, walk over to the car and check for continuity of the battery terminals and wires between the fuse box,starter, and alternator.
If they checkout, put your battery back in and start the car. as soon as you start, check the voltage at the battery again. If it reads 13-14 volts your ALT is good. If it still reads 12.6ish volts your ALT isn't charging the battery, which means it may not be powering the ignition/ecu/etc.,but the battery is. And yes a car can run without the alt working, until your battery becomes discharged. Dont believe me, if you ever decide to buy a new battery and your old one is still good, remove the belt from the alt and start the car, Sit there and stare at your car until it turns off, trust me, youll give up watching it and go watch paint dry instead.