The phase behavior of fluid mixtures of hard uniaxial ellipsoids with elongations e and 1/e, and equal molecular volume, has been studied using constant-pressure Gibbs ensemble Monte Carlo simulations for e= 15 and e=20. Four distinct phases are observed: isotropic (I), uniaxial nematic (N+ and N_) and biaxial nematic (B). The region of stability of the biaxial phase is found to be limited severely by demixing into two coexisting uniaxial phases. This is in agreement with recent theoretical predictions. The theory, however, does not account for the surprising asymmetry of the phase diagram that we find in our simulations.