Evidence for nonthermal activity in clusters of galaxies is well established from radio observations of synchrotron emission by relativistic electrons. New windows in the extreme-ultraviolet and hard X-ray ranges have provided more powerful tools for the investigation of this phenomenon. Detections of hard X-rays in the 20Y100 keV range have been reported for several clusters of galaxies, notably from Coma and others. Based on these earlier observations we identified the relatively high-redshift cluster 1ES 0657À558 (also known as RX J0658À5557) as a good candidate for hard X-ray observations. This cluster, also known as the Bullet Cluster, has many other interesting and unusual features, most notably that it is undergoing a merger, clearly visible in the X-ray images. Here we present results from successful RXTE observations of this cluster. We summarize past observations and their theoretical interpretation that guided us in the selection process. We describe the new observations and present the constraints we can set on the flux and spectrum of the hard X-rays. Finally, we discuss the constraints one can set on the characteristics of accelerated electrons that produce the hard X-rays and the radio radiation.