Even without geothermal activity, the sheer pressure of a mile deep ice sheet would keep the bottom portion in a liquid state. This is nothing new. See Lake Vostok.
I'm sorry but that is not accurate. Water under pressure will still freeze but the freezing point will be lowered only slightly. The fact that the bottom of Lake Vostok is unfrozen is due entirely to geothermal activity..........An analogy would be why you have to dig fence poles at a certain depth below your prevailing frost lines.
As a side note, the deepest known ice lies in the antarctic and rests 2,555 meters below sea level, where the ice is over 4 kilometers thick.