We report concurrent radio and gamma-ray observations of PSR B1509-58
carried out by the Parkes Radio Telescope and by the Burst and Transie
nt Source Experiment (BATSE) and the Oriented Scintillation Spectromet
er Experiment (OSSE) on the Compton Gamma-Ray Observatory (CGRO). Gamm
a-ray light curves fitted at several energies between approximately 20
and 500 keV yield a phase offset with respect to the radio pulse that
is independent of energy, with an average value 0.32 +/- 0.02. Althou
gh this value is larger by 0.07 than that reported by Kawai et al., th
e difference is not statistically significant (only approximately 2 si
mga) when account is taken of the uncertainty associated with their re
sult. We briefly discuss the possibility that the energy independence
of the gamma-ray pulse phase is a signature of nonthermal radiation in
the X-ray/gamma-ray range and the suggestion of a dependence of pulsa
r radio-gamma-ray phase offset on pulse period.