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We report on the first firm detection of pulsed γ-ray emission from PSR B1509-58 in the 0.75-30MeV energy range in CGRO COMPTEL data collected over more than 6 years. The modulation significance in the 0.75-30MeV pulse-phase distribution is 5.4σ and the lightcurve is similar to the lightcurves found earlier between 0.7 and 700keV: a single broad asymmetric pulse reaching its maximum 0.38±0.03 in phase after the radio peak, compared to the offset of 0.30 found in the CGRO BATSE soft gamma-ray data, and 0.27±0.01 for RXTE (2-16keV), compatible with ASCA (0.7-2.2keV). Analysis in narrower energy windows shows that the single broad pulse is significantly detected up to ∼10MeV. Above 10MeV we do detect marginally significant (2.1σ) modulation with an indication for the broad pulse. However, imaging analysis shows the presence of a strong 5.6σ source at the position of the pulsar. To investigate this further, we have also analysed contemporaneous CGRO EGRET data (>30MeV) collected over a nearly 4 year period. In the 30-100MeV energy window, adjacent to the COMPTEL 10-30MeV range, a 4.4σ source can be attributed to PSR B1509-58. Timing analysis in this energy window yields an insignificant signal of 1.1σ, but with a shape somewhat similar to that of the COMPTEL 10-30MeV lightcurve. Combining the two pulse-phase distributions results in a suggestive double-peaked pulsed signal above the background level estimated in the spatial analyses, with one broad peak near phase 0.38 (aligned with the pulse observed at lower energies) and a second narrower peak near phase 0.85, which is absent for energies below 10MeV. The modulation significance is, however, only 2.3σ and needs confirmation. Spectral analysis based on the excess counts in the broad pulse of the lightcurve shows that extrapolation of the OSSE power-law spectral fit with index -1.68 describes our data well up to 10MeV. Above 10MeV the spectrum breaks abruptly. The precise location of the break/bend between 10 and 30MeV depends on the interpretation of the structure in the lightcurve measured by COMPTEL and EGRET above 10MeV. Such a break in the spectrum of PSR B1509-58 has recently been interpreted in the framework of polar cap models for the explanation of gamma-ray pulsars, as a signature of the exotic photon splitting process in the strong magnetic field of PSR B1509-58. For that interpretation our new spectrum constrains the co-latitude to ∼2°, close to the ``classical'' radius of the polar cap. In the case of an outer-gap scenario, our spectrum requires a dominant synchrotron component.