Water adsorbed on clean Rh(111) forms an ordered structure with a (root 3x root 3)R30 degrees diffraction pattern. This is facilitated by the close match of surface lattice constants for Rh(111) and the (0001) face of hexagonal ice, I-h. The preadsorption of small quantities of disordered oxygen improves the long-range ordering of the water overlayer. When a well-ordered half-monolayer of oxygen is grown on the Rh(111) prior to H2O exposure, there is no evidence of any long-range ordering of the water. However, when H2O is adsorbed on a (1x1)-O/Rh(111) surface, where there is a well-ordered monolayer of adsorbed oxygen, the adsorbed H2O forms a new high-density structure exhibiting a (1x1) diffraction pattern. The adsorbed H2O structure is epitaxial with respect to the underlying oxygen and rhodium. This structure persists for many layers of adsorbed water. On the clean Rh(111) surface, water molecules are adsorbed through the oxygen lone pair orbital. When the surface is fully covered with oxygen, the first layer of water can hydrogen bond to the surface, i.e., they likely adsorb with one or both of the hydrogen atoms pointing toward the surface. This creates a template for a novel structure that forms at low pressure, producing a high-density crystalline form of interfacial ice. This discovery suggests that other molecules, especially those that hydrogen bond, may form new structures on metals covered with a high-density oxygen overlayer, with associated consequences for interfacial chemistry.