The unusual Be binary LSI +61°303 was observed by the ASCA X-ray satellite twice during the 26 day orbital-outburst cycle. Here we present the results of the spectral analysis of the ASCA data. The emission spectrum can be characterized by an absorbed power law and is inconsistent with other single-component models such as blackbody or hot plasma emission. The data show both flux and spectral variability between the two observations. The column density is determined for LSI +61°303 , and we put upper limits on Fe line emission and Fe absorption edge optical depth. The power-law index is similar to that seen for classical X-ray pulsars; however, the low X-ray luminosity and the power-law emission in other wavebands indicate that the emission is not from an optically thick accretion column but rather synchrotron or inverse-Compton emission from relativistic particles. We argue that the sub-eV emission is synchrotron, the super-eV emission is inverse Compton and that the magnetic field in the emission region is a few hundred gauss.