Heteroepitaxy of single-crystal Gd2O3 on GaN (with a wurtzite hexagonal close-packed (hcp) structure) and single-crystal GaN on Gd2O3 was achieved. In situ reflection high-energy electron diffraction reveals a sixfold symmetry in the in-plane epitaxy of the rare earth oxide on GaN and also in the overgrowth of GaN on the oxide. Single-crystal x-ray diffraction measurements find that these single-crystal oxide films are indeed the high temperature hexagonal phases of the sesquioxides, stabilized by the GaN substrate epitaxy. Despite a large mismatch in the lattice constant, the fully relaxed oxide films are of excellent structural quality. The x-ray diffraction results revealed that the GaN grown on the rare earth oxide is a single-crystal and has the same crystallographic hcp structure as the underlying GaN. The structures of both layers of GaN were also studied by cross section transmission electron microscopy.