In order to keep the density of water at 1kg/litre as you decrease its temperature below 0 centigrade you would need a container that remained rigid at many thousands of atmospheres pressure. The water would then remain liquid until something like -20 degrees centigrade.
At still lower temperatures it would become solid, but it would not form crystalline ice which always has a density lower than 1kg/litre. Instead it would form a glassy solid known as amorphous ice which has a higher density under such pressures.
There are several high density phases of amorphous ice and to get the exact details of which phase it would pass through you need a phase diagram showing density as a function of temperature and pressure extending to very high pressures and low temperatures. You can then follow the contour corresponding to a density of 1kg/litre to see how the pressure and phase would vary as the temperature decreased. I can only find enough information to give the vague answer above and it is possible that the diagram is not known well enough yet to say much more.