A computer simulation study of the phase behavior of the dipolar Gay-Berne liquid crystal model is presented. The phase transitions are determined with isothermal-isobaric (NPT) Monte Carlo simulations, utilizing the reaction field method. The electrostatic forces are found to have a considerable effect on the nature of the observed phases, but the density at which the isotropic fluid becomes unstable with respect to partially ordered phases is seen to be remarkably insensitive to the strength of the dipole. We pay particular attention to the structure of the mesophases, combining information from several singlet and pair distribution functions to build up an accurate picture of the molecular arrangement of the systems.