The growth of an intact water monolayer on Ru(0001) has been investigated by comparing the ordering of O and Ru, determined by low-energy electron diffraction (LEED), with that of the top layer of O and H, as probed by He atom scattering (HAS). Although LEED shows that water forms an extended commensurate (square root 3 x square root 3) R30 degrees structure as the coverage approaches 0.67 monolayer, the HAS distributions are insensitive to the exact water coverage and show a very low specular reflectivity, indicating a disordered water layer. The angular profile from a D2O monolayer shows a broad diffuse peak in the angular scattering distribution at a momentum exchange similar to the position of the second-order (1/3, 1/3) peaks, but the maxima show little variation with scattering azimuth. H2O shows a slightly higher He reflectivity and more clearly resolved angular structure, with broad, faint peaks appearing close to the first-order diffraction positions. The origin of this disorder is discussed based on density functional calculations for the monolayer which find that water forms chains of flat and H-down molecules within a hexagonal hydrogen-bonding network, rather than the ice bilayer usually assumed. This arrangement leads to long-range order in the O location, but disorder in the O height and the proton orientation. We discuss how this combination of lateral order in the adsorption site, but disorder in the water orientation, is reflected in the sharp square root 3 LEED pattern but diffuse, broad peaks in He scattering.