The Coma Cluster was observed in 1996 for similar or equal to 90 ks by the PCA and HEXTE instruments aboard the Rossi X-mv Timing Explorer (RXTE) satellite-the first simultaneous, pointing measurement of Coma in the broad, 2-250 keV energy band. The high sensitivity achieved during this long observation allows precise determination of the spectrum. Our analysis of the measurements clearly indicates that in addition to the main thermal emission from hot intracluster gas at kT similar or equal to 7.5 keV. a second spectral component is required to best fit the data. If thermal, this component has a temperature similar or equal to 4.7 keV, and it contributes similar or equal to 20% of the total flux. Alternatively, the second component can be a power law, likely due to Compton scattering of relativistic electrons by the cosmic microwave background. This interpretation is based on the measurements of diffuse radio emission and the similar values of the radio and X-ray spectral indices. A Compton origin of the nonthermal component would imply that the volume-averaged magnetic field in the central region of Coma is B similar or equal to 0.2 mu G, a value that is free of the usual assumption of energy equipartition. The energy density of the emitting electrons would then be similar to 8 x 10(-14) ergs cm(-3). Barring the presence of unknown systematic errors in the RXTE source or background measurements, our spectral analysis yields considerable evidence for Compton X-ray emission in the Coma Cluster.