The OSSE experiment on the Compton Observatory observed SN 1993J during three intervals, approximately 9--15, 23--36, and 93--121 days after outburst. There is evidence for continuum emission below 200 keV in the first two of these periods. Power-law fits yield intensities at 100 keV of (1.82+/-0.39)*E(-3) photons cm(-2) s(-1) MeV(-1) and (0.89+/-0.35)*E(-3) photons cm(-2) s(-1) MeV(-1) , and photon indices of -2.3+/-0.5 and -2.2+/-0.9, respectively. There is no evidence for any emission in the longer, more sensitive, third observation. These continua are too bright and too steep to be entirely due to Comptonized gamma-rays from radioactive (56) Ni and (56) Co alone. A thermal bremsstrahlung spectrum, for example, also adequately describes the OSSE data, with kT =~ 75 keV. These continua extrapolate well above nearly contemporaneous measurements at lower energies. Instead, a power-law of fixed photon index -1.2 fit to the first OSSE observation extrapolates approximately to the total luminosity measured by ASCA (Tanaka IAU Circ. 5753) from 1--10 keV, one day earlier. For a thermal spectrum a higher temperature, near 200 keV, can also fit both data sets---but only marginally. This emission cannot be unambiguously attributed to SN 1993J. Because of the large OSSE field of view, SN 1993J cannot be separated from other sources such as the nucleus of M81 or even M82. However, OSSE did observe this region twice earlier, 597 and 443 days before SN 1993J and no continuum emission was detected at either time. The apparent decline of the emission does seem to correlate well with those of SN 1993J as seen by ASCA and ROSAT. No evidence for line emission is seen in any observation. This work is supported by NASA DPR S-10987C.