Adsorption of H2O on the BaF2(111) surface was investigated at temperatures from -1 to 25 degreesC using infrared spectroscopy. At coverages near a monolayer, adsorbed films have spectra that are strikingly similar to those theoretically predicted for the surface bilayer of I-h ice. At multilayer coverages the spectra become similar to those of bulk liquid water. The coverage dependent values for enthalpy and entropy of H2O adsorption were derived from adsorption isotherm measurements. The enthalpy of formation of the monolayer is greater than that for water condensation to either the liquid or solid phases. The absolute entropy of the monolayer reveals a high degree of order having a value near that expected for ice if it were to exist at room temperature. Both the enthalpy and entropy approach the values for bulk liquid water for increases to multilayer coverages. Our results are consistent with an icelike hexagonal H2O adlayer on BaF2(111) at surface temperatures far above the ice melting point, if the adlayer thickness does not exceed one monolayer. The structure of the H2O adlayers at higher coverages is also discussed.