Travel times and amplitudes of PKP and PKKP from three earthquakes and four underground nuclear explosions are presented. Observations of reflected core waves at nearly normal angles of incidence provide new constraints on the average velocities in the inner and outer core. Interpretation of these data suggests that several small but significant changes to Bolt's (1962) core velocity model (T2) are necessary. A revised velocity model KOR5 is given together with the derived travel times that are consistent with the 1968 tables for P. Model KOR5 possesses a velocity in the transition zone which is per cent lower than that in model T2. In addition, KOR5 has a velocity jump at the transition zone boundary (r = 1782 km) of 0.013 km/sec and a jump at the inner core boundary (r = 1213 km) of 0.6 km/sec. These values are, respectively, 1/20 and 2/3 of the corresponding model T2 values.