It would evaporate as you suggest, but not instantly. That is the basis of "freeze drying" or lyophilisation. The drying speed is limited by the rate of energy supply; if you were to keep your ice in the shadow of Pluto where there is little heat and little radiant energy, it would evaporate extremely slowly. In fact, a large "wet" comet (containing lots of water in the form of ice only, and very dry indeed) might take billions of years to lose all its water and even all its ammonia.

Absolute zero simply does not nearly get a look-in, except possibly with hydrogen or helium. But I doubt that we have any place outside a lab in our solar system where they liquefy.

The rate of evaporation generally balances the rate of energy input and the vapour pressure (which in space is negligible). The faster a substance evaporates, the more it cools, until the cooling brings it down to a temperature where it evaporates at just the rate to maintain a steady temperature. On the moon it would evaporate rapidly in the sun, but slowly during the lunar night. Impurities in the water, such as dissolved salts, would settle out as a fine powder.

But at absolute zero, all things will 'freeze' so become solid.

All things will freeze well above absolute zero, exept

helium, but  helium  will not freeze at all!

Nevertheless my impression is, You believe that "frozen"

things do not evaporate?

Regarding temperatures on the moon, here are very deep thoughs

by Christian Morgenstern on this topic:

Der Mondberg-Uhu hat ein Bein,

sein linkes Bein, im Sonnenschein.

Das rechte Bein jedoch des Vogels

bewohnt das Schattenreich des Kogels.

Bis hundertfunfzig Grad im Licht

gibt Herschel ihm (zwar Langley nicht),

im Dustern andrerseits desgleichen

dasselbe mit dem Minuszeichen.