The dielectric properties of alpha-MgH2 are investigated in the photon energy range between 1 and 6.5 eV. For this purpose, a sample configuration and experimental setup are developed that allow both optical transmission and ellipsometric measurements of a transparent thin film in equilibrium with hydrogen. We show that alpha-MgH2 is a transparent, color neutral insulator with a band gap of 5.6+/-0.1 eV. It has an intrinsic transparency of about 80% over the whole visible spectrum. The dielectric function found in this work confirms very recent band-structure calculations using the GW approximation by Alford and Chou (unpublished). As Pd is used as a cap layer we report also the optical properties of PdHx thin films.