The answer to this question is going to depend on a number of variables that were not mentioned here. You did not mention exactly what kind or size of container or the amount of water placed in the container or the method of freezing. In general, when we place water in a container and freeze it, if there is no room for expansion, the container will bust. This happens regardless of whether the container is plastic/PVC or something more rigid like iron. At normal freezing temperatures, 32 degrees F, and under normal rates of pressure, water will expand about 9 percent when frozen.

However, at lower temperatures and with higher rates of pressure, there are other types of ice that are formed that do not require any expansion whatsoever. Ice formed under high pressure has a different molecular structure and density than the kind of ice that is formed in your freezer at home or in nature. Scientists have readily identified at least 15 different types of ice, or ice phases. If you were to take a stainless steel pipe that is closed off at each end and place a small amount of water in the pipe, then freeze it rapidly by pouring liquid nitrogen on it, the water would form ice 1 and then progress through the other stages of ice as the pressure continues to build and the temperature continues to drop. As the water freezes, the pressure inside of the container will continue to build. Provided that the container that you are using is rigid enough to resist the increasing pressure, the ice will continue to take on new forms and progress through the other phases.

These different phases, however, do not imply that the ice reverts back to a liquid form, at least not entirely. At the triple point of ice, which is known as Ice 2, the ice can take on three different forms at one time, with water, ice, and water vapor coexisting. If you were to rapidly freeze water in a rigid enough container (like what was describe above), this state, or ice 2, is what you would achieve....provided that the pipe or container did not rupture prior to achieving this.

Ice 2 is achieved when the temperature reaches 273.16 K (32 F) and the pressure is 611.73 Pa ( 0.088 723 935 471 psi). We know that ice melting under pressure is the reason for glacier movement, and that amorphous ice, which is found in outer space, is very different in structure than ice formed here on earth.

"What happens to water is it is allowed to freeze but prevented from expanding?"

I KNOW! I KNOW!

It becomes very cold. Yay! I win!

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Actually it depends on the method (which includes rate) of cooling. But the generalized answer I'll offer is it becomes "an amorphous solid water".

It is interesting to note that the site linked below documents 8 types of ice formation. Atmospheric pressures and method of cooling determine how and if these types form and in what relationship (or order) they develop next to other types.

http://www.benbest.com/cryonics/iceform.gif

The container suggested in the question would alter the atmospheric pressures as the temperature dropped which would change the progression based on how quickly the temperature was dropping and other factors.

It would have to be a robust container since at -7.6 degrees Fahrenheit, ice can exert pressures of ~40,000 lbs. per square inch without melting. The conclusion I offered is that if water that is freezing is unable to expand, it remains a liquid.

See also, "vitrification".

Another answer would be that it actually never technically freezes so the question as asked negates the possibility of a solid answer. (Pun intended.)

The confusing part is that you are assuming the freezing temperature of the water would be constant. As the water began to freeze and expand, the pressure inside the container would increase. The increased pressure would in turn lower the freezing point of the remaining water.

So in the end one of two things would happen:
1. The freezing point would be lowered enough that most of the water would just remain liquid.
2. If it was cold enough the water would break the container.

This video illustrates the second container pretty amazingly:
http://www.youtube.com/watch?v=G-Ywwy__pxo

According to Greg Bradburn of newton.dep.anl.gov, he posted that "The answer is -- the water pipes break when the water freezes in them. The freezing point of the water in the sphere would be depressed a small amount so at the standard (atmospheric) pressure freezing point of water the water in the sphere would remain liquid. But, at just a slightly lower temperature it would freeze and the sphere would burst."

Truthfully, I don't know. First, you'd need an "ideal" rigid container. If you use steel it will still expand, or it will burst. The amount of pressure exerted by freezing water that is being compressed is unbelievably high. If I had to speculate, I'd say that if you were able to exert enough pressure, and cool the water down at the same time sufficiently, it would freeze. However, it might require a much lower temperature to freeze the water than 0 degrees C.