Clusters of galaxies are believed to be dominated by dark matter. Some of this matter is presumably bound to galaxies in the form of massive halos, while the rest moves freely in the cluster potential well. The exact fraction of dark matter bound to galaxies is an important datum for models of cluster evolution, since time scales for orbital decay, merging, stripping, etc. are sensitive functions of galaxy mass. In this study we attempt to put a firm upper limit on the amount of dark matter associated with galaxies in clusters, by calculating the response of a galaxy with an initially massive halo to the mean tidal field produced by the overall cluster potential well. If the velocity dispersions of galactic halos are roughly equal to those of luminous galaxies, σg, it is easy to show that the truncated mass of a spherical galaxy orbiting near the center of a cluster is roughly mg ≈ G−1σg3σc−1Rc ≈ 4 × 1011M⊙, where σc and Rc are the cluster velocity dispersion and core radius. The precise value of mg must depend on the orbital geometry, as well as the number of pericenter passages since cluster formation, among other factors.