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The most common reason for a ground wire to burn up, is it's usually asked to carry too much load because the main ground wire is not connected or has failed for some other reason.
I don't know exactly what you are working on, so I can't really say. The first statement I made was a general one for lots of cases. What brand is ths car alarm, and what exactly is it's function(blow the horn)? Do they have a website with some info and diagrams? Did you take the unit apart to find the LED problem?
Actually, I let the smoke out of the LED (emanated from behind the dash) and upon inspection, just the neg. wire TO the LED itself (has it's own ground) had burned up. Everything else is in tact and operational...