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First let me say I hate troubleshooting electrical problems. Ok that having been said, my issue is that while the engine is running the battery light is on. I have checked the alternator and its output is fine. I have checked the fusable links and they appear fine. The starter has been replaced and has no problems with overheating. The cables have nominal resistance.
The battery will run down under normal use in about a week unless I am using the lights.
My question is this: Where should I start looking now?
You said you checked your alternator out and it checks fine. How did you check it out? Did you put a meter on your battery with the engine running? What was the voltage? You should be looking for somewhere around 13.2 to possibly 13.8 volts. I have made the mistake of carrying my alternators to Autozone and their test equipment always tell me my alternators were good and I ended up carrying it to a regular automotive alternator and starter place who have the proper equipment to check it out properly. They may charge you a couple of dollars but it will be worth the outcome.
You mentioned quote: The battery will run down under normal use in about a week unless I am using the lights. You mean to say if you are running the car with the lights on the battery will not go dead? It sounds like your alternator is operation normally. When you say battery light you do mean the Gen light correct? What is the reading on your voltmeter? Possibly bad battery or possibly the battery negative ground
If the "battery" light is glowing whenever the engine is running, there is a problem. Does it go out when the car is at speed (30 mph or more)? If the light ever goes out, then the alternator is weak or has one [or more] of the (three) diodes failed. If that light is always on, your problem might be the alternator, battery, or some kind of grounding fault. Call someone in a local Corvette club and ask who they might recommend to check it out. [It could be that the alternator provides just over 12v to the battery when at elevated RPM, and allows the battery to stay close to required voltage....unless you are in stop-go traffic with the A/C on. Then it can't keep up.]
I had the same problem over a year ago. With the light on and my battery losing ground even when driving regularly. The fix turned out to be fairly simple. Apparently since the battery is contained in the enclosed box behind the seat the wiring can deteriorate supposedly from being in close proximity to the battery with its acid. In any event, once the wiring inside the battery compartment was replaced the problem was fixed.
So, my suggestion would be to check the wiring and all connections within the battery compartment. I don't remember that they even looked to be in particularly bad shape but replacing them made all the difference.