The ice melts from the ice maker forming icicles

Just had this same issue repaired yesterday, the defrost element was out allowing the freezer to freeze up stopping the fan which moves the air and in turn causing the upper part of the frezzer to become warm and melt the ice. This is a common problem for the GE profile and I recommend finding a service company cheaper than GE and quit letting GE make more money off their CRAPPY appliances.

could have a problem with fill hose,if it has a **** going down the middle the water is spraying and overflowing out the ice maker,if that's ok check to see if there is an addjustment screw on the ice maker