How does freezing water to make ice remove whatever salts were in the water to begin with?
Answer
In simple terms, there isn't any space in the ice crystal lattice for the extra atoms and there is no way to plug either of the ions (or the whole salt molecule) into the growing pattern.
So more and more water joins the frozen mass, leaving a more and more concentrated brine until essentially all the water is frozen and the salt remains behind. As Manishearth notes in the comments this requires getting things rather colder than the usual "freezing point" of water.
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