Per Chemomechanics’ Answer, to -22C:
It will cool as water at room pressure and temperature and volume (and hence room density) to 0C. Then as it is cooled it goes from 0C water to 0C ice-water at around 10 atmospheres. Then it will cool to -22C as a mixture of water and ice, and it will take over 2,000 atmospheres of pressure to keep it at same volume/density. At -22C it finally becomes all ice.
Below Chemomechanics’ Answer Down to 0K:
Then it will cool as a combination of regular ice and ice-III (tetragonal crystalline ice) (somewhere around half each) to -38C, quite surprisingly staying at about that same pressure (as it cools not much more pressure above the 2,000 atmospheres is needed to hold its volume constant). Then it becomes a combination of regular ice and ice-II (rhombohedral crystalline form of ice with a highly ordered structure, also somewhere around half each), still at about the same pressure. Finally, below ~165K, the combo is regular ice and ice-IX, and this cools to 0 K, surprisingly again at around the same pressure (~2000 atmospheres).
So: From 0C water to 0C ice/water combination, at the same density, pressure goes from 1 atmosphere to over 10. Then from 0C to -22C, pressure increases to 2,000 atmospheres and it finally becomes all ice at that point. Then pressure doesn't increase much all the way to absolute zero.
How I determined this and more detail
A large metal ball can still deform locally, even if infinite radius. It wouldn’t burst but it would change shape, because the force is large.
That said we can imagine cooling water holding volume constant. If you have a constant mass of H2O and a constant volume then you have a constant density. And a constant “specific volume”, which is 1 / density.
Cooling to -22C:
I will refer to the expert Chemomechanics’ long answer above with pics and explain it for a layman, as far as it goes (it goes to -22C, 210Mpa, which is the first point of 100% ice, and then stops). Then I’ll answer the rest.
Look at his answer and find this text and the pic above it: “From this, we can predict the equilibrium response when cooling water at constant volume. We find that at constant volume (moving vertically downward from 0°C and 1 g/cc), over 200 MPa and 20°C undercooling is predicted to be required even to get about a 50% slush of water and ice.”
What he’s saying is that if we had 1g of water and hold the volume constant at 1 cc, then we have a density if 1 g/cc and a specific volume of 1 cc/g (or any amount of water held at that density), and that is the density of water at room temp and pressure. That’s why that particular specific volume is what our problem is. Temp and pressure can change but not specific volume. The figure says kg/cc, and then adds “times 10^(-3)”; would be simpler to just say g/cc.
So if we start at room temp and pressure of 30C it is water at 1 cc/g. That point would be above what the image covers. And as we cool and go straight down, above what what the image covers, density remains constant at 1 and so does the pressure until we get to 0C, and that point is on the image, at the top. It is where water and ice meet and freezing starts. Point coordinates (1, 0C).
Now we have two paths to imagine from that point (1 cc/g, 0C):
Hold pressure, not volume, constant and cool it. This is the normal case. That would be moving horizontally to the right. The density decreases (it expands) and goes through ice-water and becomes ice at 0C and ~1.08 cc/g. Temperature doesn’t decrease even when cooling until it is all ice. Then further cooling at room pressure would make the ice lower temp.
Our problem: hold specific volume constant at 1 and cool it. This means going straight down. And to reduce temperature even just 20C requires 200Mpa of pressure! and it wouldn’t even be ice yet – a mixture. That’s 2,000 times atmospheric pressure to hold the density (volume) constant. And that is enough to permanently deform any metal so it wouldn't happen that way. You could use silicon nitride, it wouldn't give much at all (any real material would give some, but we proceed pretending it can be done).
Then he’s saying continued cooling would increase pressure more, up to 209.9 MPa, and it would be all ice then, at -22C (251K). Then he ends his answer.
From -22C to 0K:
Then surprisingly the pressure doesn’t increase much to hold that density. In the second figure we go straight down at a specific volume of 10, which is our situation in different units (note the little 10^(-4) in the lower left). It goes straight through the middle of the region which is the mixture of regular ice and ice-III (tetragonal crystalline ice) and then through the middle of the region of regular ice and ice-II (rhombohedral crystalline form of ice with a highly ordered structure). Now see third figure. Finally, below ~165K, the combo is regular ice and ice-IX.
In the third figure, all of this below -22C is the line between those phases, and it is horizontal!!! at around 200Mpa all the way to absolute zero, meaning approximately constant pressure. How do we know we stay on that horizontal line and don’t go into pure ice-II or a combo of ice-II and ice-IX? Because the densities of II, II, IX are well below 1. So it takes the combination of ice-IL or ice-IH with its higher density (lower specific volume) and one of the other ices to stay at density of 1. Meaning we stay on that horizontal line in the third figure.
Below 65K, the regular ice portion changes to ice-XI, which also has a density below 1 and is really just a different form of ice-IH. The density of all of these are (very roughly) the same distance from 1 (whether above or below) so the phase mixture is (roughly) around half each in every case on the way down.