  We study the Aharonov-Bohm oscillations of a charged particle on a ring of
radius R coupled to a dirty metal environment. With Monte-Carlo methods we
evaluate the curvature of these oscillations which has the form 1/M*R^2, where
M* is an effective mass. We find that at low temperatures T the curvature
approaches at large R>l an R independent M*>M, where l is the mean free path in
the metal. This behavior is also consistent with perturbation theory in the
particle - metal coupling parameter. At finite temperature T we identify
dephasing lengths that scale as T^{-1} at R>l and as T^{-1/4} at R<<l.
