Franny, there are two switches mounted to a module in the ice and water despensor. It sounds like one of the plastic retainers on either the levers or the module itself has broken and the water switch is being made. You will have to remove the outside cover to get at these switches and see what is causing it to remain on. To remove the cover there should be two slots at the bottom that a flat blade screw driver will fit in. Gently pry the cover outward and when it releases, lift up to remove completely. You will now be able to see the module holding the switches. If you think this might be too much of a job, turn the water off behind the refer, shut the ice-maker off by lifting the bailing arm and call a tech. Catriver...post back.
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