The results of a spectral analysis, using XMM-Newton and Chandra data of the brightest ultraluminous X-ray source in the nearby galaxy M82, are presented. The spectrum of M82 X-1 was found to be unusually hard (photon spectral index Gamma approximate to 1) with a sharp cutoff at approximate to 6 keV. The disk blackbody emission model requires a nonphysically high temperature. Instead, the spectrum is better described, with a lower reduced chi(2), as emission 2 due to the nearly saturated Comptonization of photons in an optically thick (tau approximate to 10-30, depending on the geometry) plasma having a temperature kT approximate to 2 keV. This is in contrast to the high- energy spectra of other black hole systems, which are relatively steeper (Gamma > 1.5) and hence are modeled as the unsaturated thermal and/or nonthermal Comptonization of soft photons, in an optically thin (tau approximate to 1) high-temperature plasma. An iron line emission that is marginally resolved (sigma similar to 0.2 keV) is required to fit the data. We argue that the standard geometry for the X-ray-producing region, which consists of an optically thin inner disk or a uniform/patchy corona on top of a cold disk, is not applicable to this source. Alternatively, the geometry of the X-ray-producing region could be a large sphere surrounding a cold accretion disk or an optically thick inner disk region that cools by bremsstrahlung self-Comptonization. For the latter scenario, such an inner disk region, whose effective optical depth to absorption is less than unity, is expected in the standard accretion disk theory for near-Eddington accretion rates.