I could be wrong about this (it's been awhile since I took chemistry), but I think the ice has to be floating for the Archimedes principle to apply.

You can fill a glass of ice-water right up to the brim, and it won't spill over as the ice melts. But only if the ice is floating in the water. It's because the ice's mass pushes down on the liquid water, displacing a fixed amount relative to the weight of the ice.