The authors report on the growth of single-crystal Al2O3 thin films on Nb (110) surfaces. Niobium is grown on α-Al2O3 (1120), followed by the evaporation of Al in an O2 background. Initially, Al2O3 grows layer by layer with hexagonal symmetry indicating either α-Al2O3 (0001) or γ-Al2O3 (111). Diffraction measurements show that the Al2O3 initially grows clamped to the Nb with tensile strain near 10%. This strain relaxes with further deposition and beyond about 50 Å, the authors observe island growth. Despite the asymmetric misfit between Al2O3 and Nb, the strain is surprisingly isotropic. Josephson junctions employing epitaxial Al2O3 show low effective tunnel barriers and high leakage currents.