We have demonstrated experimentally the existence of inertial waves in a slowly precessing spheroid of fluid. Although such oscillatory internal shear layers have been predicted theoretically and numerically, previous precession experiments had shown no evidence of their presence. Using an ultrasonic Doppler velocimetry technique, profiles of radial velocity have been measured in our precession experiment. Comparison of these profiles with their synthetic counterparts obtained numerically, proves the presence of the predicted internal shear layers. They are emitted from the breakdown of the Ekman layer at the two critical latitudes of the fluid (around 30¿ and -30¿) and propagate through the entire volume on conical surfaces. The asymptotic laws for these oscillatory layers, confirmed experimentally and numerically, lead us to predict an oscillatory flow of 10-6 m/s along such characteristic cones in the Earth's fluid outer core. ¿ 2001 American Geophysical Union