The highly variable BL Lacertae object Mrk 421 has been observed simultaneously in the radio, optical ultraviolet, and X-ray bands over a period of 4 days in early 1984 December and once again in early 1985 January. Using the EXOSAT observatory, we found that dramatic changes occurred in the X-ray flux on a time scale of less than a mouth. During this time the 2-10 keV flux dropped by a factor of 8, whereas the 0.1-1 keV flux decreased by a factor of only 2. These changes were not reproduced at longer wavelengths during the period of simultaneous observations. However, a drop in the ultraviolet flux occurred some months later, which is consistent with the longer characteristic loss times for the lower energy electrons. Since the ultraviolet through radio flux is stable when the X-ray flux is changing, it is extremely unlikely that a simple synchrotron model can account for the full spectrum; in this model the whole spectrum is expected to rise uniformly and in phase as a result of the injection of energetic particles. A simple synchroton self-Compton model that is self-absorbed in the radio also requires an X-ray flux which is many orders of magnitude greater than is observed. However, this discrepancy may be explained by relativistic beaming of electrons with delta > approximately 40 or by a model in which the self-absorption turnover occurs in the optical, and the synchrotron break occurs in the X-rays. Shorter time scale (approximately 10,000 s) variability was also apparent in the 2-10 keV X-ray light curves, and we suggest that it may be a direct measure of the injection time scale. Although reasonable fits resulted when the X-ray data were compared with a simple power-law model with some absorption, a substantial improvement in chi 2 was obtained by adding a high-energy exponential cutoff. Use of this model produced a spectral index close to that typically found in the optical for BL Lacertae objects, in contrast to the high values usually inferred from X-ray spectra.