  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 $1.36\pm0.01$, 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
$2.215\pm0.005$. 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.
