We demonstrate that earth nutation measurements made with very long baseline interferometry are of sufficient accuracy to be sensitive to the properties of the core-mantle boundary. The retrograde nutation with annual frequency is particularly sensitive, since this frequency is closest to that of the free core nutation, a nutational normal mode which produces relative motion of the core and mantle. Our nutation measurements imply a deviation of the amplitude of the retrograde annual nutation from its value as calculated by Wahr and by Sasao et al. If this deviation is interpreted as the effect of a departure of the core-mantle boundary from its hydrostatic figure, then the observed amplitude is consistent with a core-mantle boundary that has a second zonal harmonic deviation from the hydrostatic equilibrium figure, with the peak-to-valley deviation being 490¿110 m. The part of the retrograde annual nutation out of phase with the driving torques yields an upper limit on the kinematic viscosity at the surface of the fluid core of 0.54 m2/s (99.5% confidence limit).