The wave-vector and directional dependence of the Rayleigh elastic wave localized in epitaxial films of bcc cobalt are measured by Brillouin scattering. The Rayleigh velocities are well described by elastic continuum theory and yield values for the ${\mathrm{C}}_{11}$ and ${\mathrm{C}}_{44}$ elastic constants that are reduced to about 65% of the corresponding values in the thermodynamically stable hcp configuration. While the reduction in ${\mathrm{C}}_{11}$ in these supported films (several hundred angstroms thick) is consistent with recent calculations for bulk bcc cobalt, the ${\mathrm{C}}_{44}$ and ${\mathrm{C}}_{12}$ values we have derived are significantly smaller than those predicted by the theory. The ${\mathrm{C}}_{\mathit{i}\mathit{j}}$'s determined independently from the Rayleigh wave dispersion along distinct crystallographic directions in films of different orientations agree well.