High temporal resolution measurements of ion velocity distributions obtained with the Los Alamos/MPI instruments on ISEE-1 during a high Mach-number bow shock crossing under nearly perpendicular conditions reveal the presence of a beam of secondary ions gyrating about the magnetic field. These ions are first encountered as the satellite enters the region just ahead of the foot of the shock and show the behavior expected for solar wind ions which are reflected off the shock and subsequently returned to it under the combined influence of deflection in the magnetic field and acceleration in the interplanetary electric field. In particular, the energy and direction of the beam ions, as well as the distance from the shock ramp when they first appear, show good quantitative agreement with predictions from simple ion orbit considerations. Beam densities increase from 0.2% to 30% of the total plasma density as the main shock is approached. The agreement with predictions based on unimpeded orbits indicates that the ion beam is not measurably degraded through interaction with the counter-streaming solar wind upstream of the ramp. The significance of the reflected and gyrating ions lies in the fact that they provide strong dispersion in velocity space and thus constitute the first step in the process by which solar wind streaming energy is converted into thermal energy of the ions at the bow shock.