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Corvette Clocks by Roger are a direct quartz conversion, not hard to do, had mine for 10 years and works perfect around $100.00 http://www.corvetteclocks.com/
Use search and try to find a thread with a link to a guy's page where he fixes his clock. I used his instructions and pics to fix mine. I just cleaned it, lubricated it, and carefuly cleaned the points.
I know this may sound like "yeah, sure" but won't cost anything to try.
First off I know you have a relatively low mile original Vette.
My 8K mile 79 had a clock that didn't work for the first 22 years that I had the car.
One day just messing with it I pulled out the adjuster and set the time then snapped it back in. Did this a few times and the clock started working! Been working fine for the last 3 years. Apparently just 'stuck'.
Keep in mind the factory clock is like an old battery operated kitchen clock. Battery winds it, spring runs it; battery winds, spring runs; battery, spring.......
But actually I don't worry about the clocks. I have battery disconnects on all my Vettes.
Fuse first! Same one as interior lights and anti theft system. If you decide to remove the clock for cleaning or any other reason, be aware that the "decorative" allen heads in the cluster are exactly that! If you try to "unscrew" them, all you will so is round out the plastic holes. The cluster panel has to be gently removed and the clock taken out from the back. Console side panels (and seats, to get to them) have to come off, and the radio must be loosened from its rear mounting and from the cluster panel. Console top panel has two screws at its forward end to come out, too, or something will break. The gauges and clock all come off the panel from the back. Once out, the clock is bench work, using electronic contact cleaner/lube, and a points file.
If your anti-theft isn't working, now's the time to investigate it as well. That relay is inside the console, right next to its flasher (standard turn signal style). Hot wire the horn behind the left front fender liner to test it. If it's working, then it's usually door, hood, and T-top switches to check. I've had good luck with the '75 & '78 that I've done for both clock and alarm.
Like the other guy said, though, try winding the clock one full hour with the power engaged. That may save you all of the above. If it takes off, wind it all the way back one hour to compensate the (supposed) automatic adjustment. PM me if you have questions, good luck!