We report X-ray observations of the powerful radio galaxy Cygnus A with the Chandra X-Ray Observatory. This Letter focuses on the radio hot spots, all four of which are detected in X-rays with a very similar morphology to their radio structure. X-ray spectra have been obtained for the two brighter hot spots (A and D). Both are well described by a power law with photon index Gamma = 1.8 +/- 0.2 absorbed by the Galactic column in the direction of Cygnus A. Thermal X-ray models require too high, gas densities and may be ruled out. The images and spectra strongly support synchrotron self-Compton models of the X-ray emission, as proposed by Harris, Carilli, & Perley on the basis of ROSAT imaging observations. Such models indicate that the magnetic field in each of the brighter hot spots is 1.5 x 10(-4) G, with an uncertainty of a few tens of percent. This value is close to the equipartition field strengths assuming no protons are present. The possibility that the X-rays are synchrotron radiation is briefly discussed but not favored. We speculate that production of the gamma similar to 10(7) electrons necessary for X-ray synchrotron radiation from hot spots is inhibited when the external gas density is high, as is the case when the radio galaxy is within a cooling flow.