When control panel is in the "Water" position, a very slow leak occurs, the dispenser drip area fills with water, and seeps out to leak down the front of the refrigerator door. The leak is super slow... about one drip per 30 minutes. When the control panel is set for either ice position the leak continues for a day or so then ceases all together. I am not sure what to buy or how to take out the water ice dispenser to further diagnose the problem.

I ended up having a GE repairman come out. He said he thought it was related to condensation buildup in the door and around/behind the dispenser, and was not a leak. He said the door is filled with foam and that there is no way top replace any components inside, not even the tube. He said if it was a leak that the only thing to replace was the water valve, which is located behind the panel in the rear. I had him do that in hopes that that would fix it it. It did not. The leak continued. So, what I ended up doing was opening the panel in the rear and disconnecting the line to the door from the line out of the valve. This cut off water supply to door but lets water still go to the ice maker. That stopped the drip but also stopped water from dispensing out of the door. Ice still dispense from door. I clamped off the line out of the valve (in the rear) so that if someone put it in the water position it wouldn't dump to the floor. Hope that helps.

How Did You Get To The Line In The Door?

I have a GE Profile Stainless Steel fridge.The dispenser has a poor water containment design. With the fridge unplugged, I removed the control panel and the ice dispenser housing. 4 screws. I saw a possible problem where the water tube comes out. Condensation was dripping i to that area with a possible hole. I cleaned and sealed with silicone.
The stainless steel trim was easily removed at this point. A lot of water collected in the door insulation. If the tray overflows it collects there. I dries the area best I could. Used paper towels and stuck them in the holes and change them frequently. I was able to remove all the water marks with Brasso polish.
Next, working quickly with silicone is essential. A credit card works great in cleaning excess. I filled the holes and sealed all of the tray seams. The removable tray was set aside. I sealed around the stainless steel trim about a quarter way up. Snapped everything back and cleaned the excess silicone. I let dry and used some Shelia Shine to finish the job.