The structure and energetics of a monolayer of water on a perfect MgO substrate is investigated by ab initio calculations. Several minima in the potential energy surface were found including both physisorbed and chemisorbed states in which one third of the water molecules were dissociated. In the more stable of the physisorbed states, the water molecules were not all parallel to the surface, but some showed hydrogen bonding with surface oxygen ions. Even in the physisorbed state, the geometry of these surface H-bonded water molecules were distorted relative to the bulk. One of the stable chemisorbed states was the same as that found in earlier work by Giordano [Phys. Rev. Lett. 81, 1271 (1998)]. Two more stable chemisorbed structures were found in which strong hydrogen bonds to the hydroxide ions in the water layer were formed. The existence of different minima appears to be the result of a subtle interplay between hydrogen bonding between adsorbed species and with the surface oxide ions. Harmonic vibrational frequencies were determined for both a chemisorbed and a physisorbed states and spectroscopic features which should discriminate between chemisorbed and physisorbed states are identified.