Bill- Nice question. I'll assume you mean a very rigid container, since otherwise the answer is boring- the container just expands.
When the water reaches 0°C, a little bit starts to freeze. (Here I'm assuming that there's some dirt or something to nucleate the freezing, so the water doesn't form a supercooled liquid.) That frozen part occupies a little extra volume, so the remaining water is compressed, under higher pressure. That lowers its freezing point. As you keep lowering the temperature, more and more ice forms. Finally, at some lower temperature the whole thing is compressed ice.
In other words, at fixed volume there’s a range of temperatures at which ice and liquid coexist, instead of just one temperature. With a little effort to look things up, one could calculate how big the range is, using the volume difference between ice and water at atmospheric pressure, the compressibilities, and the latent heat of freezing.
I’m assuming that ordinary ice forms under these pressures, although at high enough pressures different types of crystal ice become stable.
(published on 10/22/2007)