The interfacial Dzyaloshinskii-Moriya interaction (DMI) in an in-plane anisotropic Pt(4nm)/Co(1.6nm)/Ni(1.6nm) film has been directly observed by Brillouin spectroscopy. It is manifested in the asymmetry of the measured magnon dispersion relation, from which the DMI 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. ; Comment: 15 pages, 4 figures