We explain the origin of the Kondo mirage seen in recent quantum corral Scanning Tunneling Microscope (STM) experiments with a scattering theory of electrons on the surfaces of metals. Our theory combined with experimental data provides the first direct observation of a single Kondo atom phase shift. The Kondo mirage observed at the empty focus of an elliptical quantum corral is shown to arise from multiple electron bounces off the corral wall adatoms in a manner analagous to the formation of a real image in optics. We demonstrate our theory with direct quantitive comparision to experimental data. *This research was supported by the National Science Foundation under Grant No. CHE9610501 and by ITAMP.