We present inelastic cold neutron scattering measurements on a single crystal of the heavy-fermion compound CeCu5.9Au0.1, where non-Fermi-liquid behavior near a quantum phase transition was found in the specific heat and resistivity. This compound shows strongly correlated magnetic fluctuations, most intense at wave vectors Q(1), near(1,0,0), close to the magnetic ordering vector found at higher Au-concentration. The energy dependence can be best described by a modified quasielastic Lorentzian with power alpha = 0.7. Down to the lowest temperature of 0.07 K the relaxation rate Gamma remains high for general Q(L)(Gamma approximate to 0.2 meV), but follows the absolute temperature with a small offset (Gamma(0) approximate to 0.02 meV) for Q(1), demonstrating that correlations are here responsible for the deviation from Fermi-liquid behavior. (C) 1998 Elsevier Science B.V. All rights reserved.