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Is there a model number on the barbeque? I can look it up with that information.
So far it sounds like a faulty flame sensor.
Locate the gas valve, it should have a larger pipe coming into to it, and the same size or larger pipe going to the burners... if it also has a 1/4" aluminum tube attached, and a spaghetti sized copper tube attached, or two wires attached it has a thermocouple, or thermopile that is bad or just needs to be cleaned off. You can buy new thermocouples or thermopiles at most hardware stores.
We can go from there.