Using synchrotron-based luminescence excitation spectroscopy in the energy range 4-20 eV at 8 K, the indirect-X optical band-gap transition in cubic boron nitride is determined as 6.36 ± 0.03 eV, and the quasi-direct band-gap energy of hexagonal boron nitride is determined as 5.96 ± 0.04 eV. The composition and structure of the materials are self-consistently established by optically detected x-ray absorption spectroscopy, and both x-ray diffraction and Raman measurements on the same samples give independent confirmation of their chemical and structural purity: together, the results are therefore considered as providing definitive measurements of the optical band-gap energies of the two materials.