The interfacial Dzyaloshinskii-Moriya interaction in an in-plane anisotropic Pt(4 nm)/Co(1.6 nm)/Ni(1.6 nm) film has been directly observed by Brillouin spectroscopy. It is manifested as the asymmetry of the measured magnon dispersion relation, from which the Dzyaloshinskii-Moriya interaction constant has been evaluated. Linewidth measurements reveal that the lifetime of the magnons is asymmetric with respect to their counter-propagating directions. The lifetime asymmetry is dependent on the magnon frequency, being more pronounced, the higher the frequency. Analytical calculations of the magnon dispersion relation and linewidth agree well with experiments.