I had the exact same problem and just fixed it! The estimate for this repair was $375.00 and the actual cost to fix it was approx $35.00 and it was incredibly easy!!! Take a thin flat head screw driver and pry off the control panel starting at the top left. Carefully loosen all the way around. It snaps in/out with no apparent lip/edge, and there are no screws holding it in place. It will stay connected by wires that run from the panel. Next remove grey plastic part which is immediately behind the control panel. It is held in place by 4 screws. Pull this out gently as it is also connected by lots of little wires.
I replaced the solenoid which totally fixed the problem. It is a small barrell shaped thing that is held in place by three screws and it is located in the upper right part of the small compartment that is exposed when you remove the grey panel. You will see that it has two little wires that run together and clip into place on the circuit board. You simple pull the plug out of the circuit board, unscrew the three screws and pull the thing out! Then reverse the process to put the new one in. The hardest part is putting the little screws back in because the space is small (a magnetic screwdriver would take care of this problem.)
I flipped the circuit breaker on my fridge so I wouldn't electricute myself somehow...and I didn't want to pull the fridge out and unplug it.
My fridge is a GE Profile model #gss25vgmdcc
Best of luck!! | 
