We report the results of molecular dynamics simulations of the structure of the rotator phase of a monolayer of a long chain alkyl carboxylic acid, H(CH2)(15)COOH, supported on the 0001 face of ice Ih, as a function of area per molecule. The potential field generated by this surface has an ordered component arising from the lattice of oxygens and an irregular component arising from the disordered distribution of hydrogens. Consequently, the many possible head group pinning sites of the ice Ih 0001 surface are irregularly distributed and not commensurate with the oxygen lattice. Our results, which are derived from simulations that use a pseudoatom model of the amphiphile molecule and a rigid lattice model of the ice surface, show that the constraint imposed by the potential field of the rigid ice Ih 0001 surface generates properties of the ice supported monolayer that are different from those of the corresponding liquid water supported monolayer. As expected, we find that the head groups are pinned and that as the area per molecule is increased the monolayer develops a collective tilt; the collective tilt increases as the area per molecule increases. More striking, we find that the projection of the unit cell of the monolayer in the plane of the surface remains, on average, hexagonal for all collective tilt angles and all surface coverages. In contrast, the introduction of a collective tilt of the molecules in the L(2) (rotator) phase of a liquid water supported monolayer is always accompanied by a distortion of the projection of the unit cell in the plane of the surface, typically from hexagonal at zero tilt to centered rectangular for all nonzero tilt. Despite this difference in packing structure, and the pinning of the head groups, the surface pressure as a function of area per molecule, and the collective tilt as a function of area per molecule of a monolayer supported on ice Ih closely resemble those for the L(2) phase of a monolayer supported at the air/water interface. The only structural data available for monolayers of long chain molecules on ice Ih refer to a low-temperature (similar to 100 K) and high surface density (similar to 18 Angstrom(2) per molecule), under which conditions the amphiphile molecules pack in a herringbone structure with a rectangular unit cell in the interface plane. The difference between the properties of this low-temperature monolayer on ice Ih, which is not in a rotator phase, and the properties of the rotator phase we have simulated is briefly discussed.