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According to the plate (https://www.dropbox.com/s/w3q5qtfj69e2orh/2013-09-10%2019.56.43.jpg) the Model is PSC23SGRB SS
I am pretty sure the black tube is the hose from the solenoid to the ice maker.
It looks like http://www.appliancepartspros.com/ge-grommet-and-tube-wr17x11168-ap3186471.html except there is no plastic at the end. It is all black no white plastic (could be broken?)
Does this help?
yes that is the water feed for the ice maker. this slides through the back of the refrigerator where I placed the arrow.its sounds like yours broke, heres a link to order a new one. http://www.repairclinic.com/PartDetail/Fill-Tube/WR17X11168/945251?modelNumber=PSC23SGRBSSif that has come out you need to put it back in .