We had a similar problem (Whirlpool French door), and we had food in the upper left corner (close to where the ice is made) which partially blocked the air flow, and allowed the water line to freeze up.
Try a Google search on your specific make and model, you will probably get several fixes. Also, I did use the hair dryer trick to unfreeze the line quickly, in around 5 to 10 minutes, and then we keep that area free, and no further problems.
If all else fails, call Beck's appliance repair, and he will be able to fix your problem.
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Pennsylvania, for 60+ years, most recently, Allentown, now TV. 
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