I think what you're saying is that maybe sugar molecules won't fit too badly into an ice crystal, so maybe some frozen state will form with the sugar in it. (There are other possible interpretations but I'm trying to find one consistent with the laws of thermodynamics. The sort of piecewise thermodynamic arguments you're thinking about are not at all reliable.)
I don't believe that mixed crystal forms, for several reasons. Perhaps the one that's least technical and most fun to describe is the story of my great-grandfather making making a sort of apple brandy by leaving hard cider out to freeze and throwing out the ice. What was left was said to be not only highly alcoholic but also quite sweet. OK, that was a slightly different solution because of the alcohol, but I still bet that slowly frozen ice will be almost pure water.
Anyway, the obvious thing to do is try the sugar water experiment before Spring or global warming ruin your beautiful NH winter. Let us know if it doesn't work.
Popsicles are essentially just frozen sugar water -- there's no real reason why sugar water won't freeze. The rate of freezing may play a very important role. Slowly freezing the water where the ice forms on one side and grows to another will probably separate the sugar out much better than quick freezing with many nucleation sites. The sugar in a popsicle may be predominantly between less sugary crystal grains. A badly frozen popsicle will probably have a sweet end and a not-so-sweet end, but it'll be very hard to get all of the sugar on one side.
Repeating the freezing process multiple times, separating the more sugary part from the less sugary part can improve the purity.
(republished on 07/25/06)