I'm not saying that you won't get a single molecule of water going into solution, but if you add a small amount of liquid to ice that's well below freezing, you

I'm not saying that you won't get a single molecule of water going into solution, but if you add a small amount of liquid to ice that's well below freezing, you really can get effectively zero melting. It will equilibrate to a temperature that's still below the freezing point of water.