We present extensive Monte Carlo simulations for the thermodynamic and structural properties of a planar bilayer of dipolar hard spheres for a wide range of densities, dipole moments, and layer separations. Expressions for the stress and pressure tensors of the bilayer system are derived. For all thermodynamic states considered, the interlayer energy is shown to be attractive and much smaller than the intralayer contribution to the energy. It vanishes at layer separations of the order of two hard sphere diameters. The normal pressure is negative and decays as a function of layer separation h as -1/ h5. Intralayer and interlayer pair distribution functions and angular correlation functions are presented. Despite the weak interlayer energy, strong positional and orientational correlations exist between particles in the two layers.