Proton order is less constrained in a fully H-bonded, adsorbed H2O monolayer than in bulk ice. Thus, though weakly, configurational entropy favors wetting by deposited H2O over formation of 3-D crystalline mounds. A Pauling-type estimate yields a 0 K adlayer entropy of similar to(1)/(2)k(B) In(9/2), almost twice that of ice Ih. Thus, at 150 K, where periodic adlayers on metals are observed, residual entropy reduces their free energies relative to a 3-D ice crystal by 4.5 meV/molecule, or similar to3% of the cost of the adlayers' broken H-bonds. A Debye model implies that vibrational entropy contributes no more than another 4.4 meV/molecule to the free energy preference for wetting, at 150 K. This information, though only a bound because of substantial uncertainty in measured adlayer Debye temperatures, is nonetheless sufficient to conclude that T = 0 K energies dominate the free energy balance between wetting and mound formation.