Liquid-solid and solid-solid phase transitions of a monolayer water confined between two parallel hydrophobic surfaces are studied by molecular dynamics simulations. The solid phase considered is the high-density rhombic monolayer ice. Based on the computed free energy surface, it is found that at a certain width of the slit nanopore, the monolayer water exhibits not only a high freezing point but also a low energy barrier to crystallization. Moreover, through analyzing the oxygen-hydrogen-oxygen angle distribution and oxygen-hydrogen radial distribution, the high-density monolayer ice is classified as either a flat ice or a puckered ice. The transition between a flat ice and a puckered ice reflects a trade-off between the water-wall interactions and the electrostatic interactions among water molecules.