Molecular dynamics simulations demonstrate that there are at least two classes of quasi-two-dimensional solid water into which liquid water confined between hydrophobic surfaces freezes spontaneously and whose hydrogen-bond networks are as fully connected as those of bulk ice. One of them is the monolayer ice and the other is the bilayer solid which takes either a crystalline or an amorphous form. Here we present the phase transformations among liquid, bilayer amorphous (or crystalline) ice, and monolayer ice phases at various thermodynamic conditions, then determine curves of melting, freezing, and solid-solid structural change on the isostress planes where temperature and intersurface distance are variable, and finally we propose a phase diagram of the confined water in the temperature-pressure-distance space.