I agree with Wavery - it sounds like a thermocouple. There is one possible though somewhat unlikely possibility. The tip of the thermocouple must be IMMERSED (that is the official term for it) in the pilot flame. In other words, it has to get really hot in order to generate enough voltage to keep the gas valve activated. If the thermocouple has been jostled out of place a little bit, or if it is covered with a thick layer of crud, then it won't get hot enough, and the gas valve will shut off.
Do you have a copy of the manual for the refrigerator? If not, perhaps you should get one on line from the manufacturer's web site. It contains pictures of how to get at the thermocouple, what it looks like, and exactly how it should be positioned in the flame.
One of our members, Commodor47, did an excellent series of posts with pictures on this exact topic a year or two ago. You should be able to find it by using the Search tool.
Bill
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