Radiation pressure on dust grains may be an important physical mechanism driving galaxy-wide superwinds in rapidly star-forming galaxies. We calculate the combined dust and gas Eddington ratio (Gamma) for the archetypal superwind of M82. By combining archival GALEX data, a standard dust model, Monte Carlo dust scattering calculations, and the Herschel map of the dust surface density distribution, the observed FUV/NUV surface brightness in the outflow constrains both the total UV luminosity escaping from the starburst along its minor axis (L_*, UV) and the flux-mean opacity, thus allowing a calculation of Gamma. We find that L_(*, UV) ~ 1-6*10^42 ergs s^-1, ~2-12 times greater than the UV luminosity observed from our line of sight. On a scale of 1-3 kpc above the plane of M82, we find that Gamma ~ 0.01 - 0.06. On smaller scales (~0.25-0.5 kpc), where the enclosed mass decreases, our calculation of L_(*, UV) implies that Gamma ~ 0.1 with factor of few uncertainties. Within the starburst itself, we estimate the single-scattering Eddington ratio to be of order unity. Thus, although radiation pressure is weak compared to gravity on kpc scales above the plane of M82, it may yet be important in launching the observed outflow. We discuss the primary uncertainties in our calculation, the sensitivity of Gamma to the dust grain size distribution, and the time evolution of the wind following M82's recent starburst episodes.