A phase diagram for a fluid comprised of hard ellipsoids and hard spheres has been calculated on the basis of a density functional theory. Needed in the theory was the direct correlation function and this was abstracted from previous equations of state of fluid mixtures of hard convex bodies. Orientational correlations in the direct correlation function were implemented only to the extent that the pair excluded volume was calculated exactly. The hard sphere, hard ellipsoid fluid mixture did not undergo phase separation in the isotropic phase. However, after the isotropic to nematic phase transition occurred, the nematic phase was enriched in hard ellipsoids. The isotropic to nematic phase transition, with its entrained phase separation, was found to be a strongly first order process.