A systematic Monte Carlo (MC) simulation and perturbation theoretical (PT) study is reported for the dielectric constant of the polarizable dipolar hard sphere (PDHS) fluid. We take the polarizability of the molecules into account in two different ways. In a continuum approach we place the permanent dipole of the molecule into a sphere of dielectric constant ∞ in the spirit of Onsager. The high frequency dielectric constant ∞ is calculated from the Clausius-Mosotti relation, while the dielectric constant of the polarizable fluid is obtained from the Kirkwood-Fröhlich equation. In the molecular approach, the polarizability is built into the model on the molecular level, which makes the interactions non-pairwise additive. Here we use Wertheim's renormalized PT method to calculate the induced dipole moment, while the dielectric constant is calculated from our recently introduced formula [22]. We also apply a series expansion for the dielectric constant both in the continuum and the molecular approach. These series expansions ensure a better agreement with simulation results. The agreement between our MC data and the PT results in the molecular approach is excellent for low to moderate dipole moments and polarizabilities. At stronger dipolar interactions ergodicity problems and anizotropic behaviour appear where simulation results become uncertain and the theoretical approach becomes invalid.