The phase diagram of fluids of hard biaxial ellipsoids with c/a 10 and b/a is an element of {1,10}, where a, b, and c are the semi-axes, has been studied using computer simulation. Four homogeneous phases are in evidence : isotropic (I), nematic (N+), discotic (N-) and biaxial (B). First-order isotropic-nematic and isotropic-discotic coexistence lines have been traced out using Gibbs-Duhem integration of the coexistence pressure with respect to the molecular biaxiality. We conclude that the isotropic-nematic transition is greatly weakened by a modest degree of molecular biaxiality, in agreement with several recent theories. The I-N+ and I-N- lines meet two second-order nematic-biaxial and discotic-biaxial lines at the Landau bicritical point. This point is predicted to occur at around the self-dual particle shape, b/a = root c/a, and so extensive simulations have been performed at and around this point. Very sluggish behaviour is expected in this region of the phase diagram and so long simulations were required. An estimate of the location of the Landau bicritical point is reported. We also highlight the asymmetry of the phase diagram about b/a = root c/a, where the ellipsoid is neither prolate nor oblate.