Water expands upon freezing. What happens when water is cooled below 0C in an undeformable, constant-volume container? This is a fundamental question in materials thermodynamics, and is also relevant in biological, geological, and technological applications in which ice forms under nano-, meso-, or macroscale confinement. Here, we analyze the phase-equilibria and kinetic behaviors of water and ice-1h in an isochoric (constant-volume) system. By making use of the Helmholtz potential
F(temperature, volume), in contrast to the Gibbs potential G(temperature, pressure), we demonstrate significant changes in phase behavior when the specific volume of the container is constrained below that of ice-1h. We construct a TV(temperaturevolume) phase diagram for water and ice that features a broad two-phase equilibrium region, and we further derive an isochoric nucleation theory that reveals the existence of a critical confinement volume, on the order of microns, below which ice-1h is kinetically prohibited from forming.