  We present the results of new Agile observations of PSR B1509-58 performed
over a period of 2.5 years following the detection obtained with a subset of
the present data. The modulation significance of the lightcurve above 30 MeV is
at a 5$\sigma$ confidence level and the lightcurve is similar to those found
earlier by Comptel up to 30 MeV: a broad asymmetric first peak reaching its
maximum 0.39 +/- 0.02 cycles after the radio peak plus a second peak at 0.94
+/- 0.03. The gamma-ray spectral energy distribution of the pulsed flux
detected by Comptel and Agile is well described by a power-law (photon index
alpha=1.87+/-0.09) with a remarkable cutoff at E_c=81 +/- 20 MeV, representing
the softest spectrum observed among gamma-ray pulsars so far. The pulsar
luminosity at E > 1 MeV is $L_{\gamma}=4.2^{+0.5}_{-0.2} \times10^{35}$ erg/s,
assuming a distance of 5.2 kpc, which implies a spin-down conversion efficiency
to gamma-rays of $\sim 0.03$. The unusual soft break in the spectrum of PSR
B1509-58 has been interpreted in the framework of polar cap models as a
signature of the exotic photon splitting process in the strong magnetic field
of this pulsar. In this interpretation our spectrum constrains the magnetic
altitude of the emission point(s) at 3 km above the neutron star surface,
implying that the attenuation may not be as strong as formerly suggested
because pair production can substitute photon splitting in regions of the
magnetosphere where the magnetic field becomes too low to sustain photon
splitting. In the case of an outer-gap scenario, or the two pole caustic model,
better constraints on the geometry of the emission would be needed from the
radio band in order to establish whether the conditions required by the models
to reproduce Agile lightcurves and spectra match the polarization measurements.
