Angle-resolved photoemission measurements of divalent hexaborides reveals a >1 eV X-point gap between the valence and conduction bands, in contradiction to the band overlap assumed in several models of their novel ferromagnetism. While the global ARPES band structure and gap size observed are consistent with the results of bulk-sensitive soft x-ray absorption and emission boron K-edge spectroscopy, the surface-sensitive photoemission measurements also show a variation with cation, surface and time of the position of the surface chemical potential in the band structure.