We report timing and spectral results for PSR B1509-58 observed by BeppoSAX in February 1998. We obtained pulse profiles with high statistical significance from 0.1 to 300 keV that exhibit the well-known broad asymmetric single pulse. The shape of this pulse does not change across this energy window and can be described as the sum of a narrow and a broader Gaussian component separated similar to 0.13 in phase. The spectral distribution can be accurately represented, over the entire BeppoSAX energy range, by a curved function rather than a simple power-law shape. The extrapolation of this model to higher energies is also consistent with the COMPTEL fluxes in the 0.75-30 MeV range, with the maximum luminosity for the broad pulse of PSR B1509-58 reached at similar to 5.4 MeV. The comparison with the Crab pulsar spectrum suggests a possible origin of the X-ray emission in terms of synchrotron radiation from secondary pair particles near the neutron star.