We present the results of observations of the PSR B1509$-$58/MSH 15$-$52 system in X-rays ($2-250$ keV) by the Rossi X-ray Timing Explorer. The spectra of the peak of the pulsed component (radio phase $0.17-0.53$) is fit by a power law of photon index $1.36pm0.01$, with no evidence of a high energy spectral break seen up to $sim200$ keV. For the off-pulse spectral component, the spectrum from $2-250$ keV is fit by a power law of photon index $2.215pm0.005$. An iron emission line at 6.7 keV with an equivalent width of 129 eV improves the fit, but only at a marginal significance. Thermal bremsstrahlung and Raymond-Smith models produce much worse fits to the unpulsed data. The lack of a high energy spectral break in the pulsed emission implies an efficiency of $geq 3%$ in the conversion of pulsar spindown energy to pulsed X-rays in the system.
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