  Using the quark-interchange model of Barnes and Swanson, we evaluate the
dissociation cross sections for $J/\psi$ and $\psi'$ by $\pi$ and $\rho$ at low
collision energies. In collisions with $\pi$ mesons near threshold the $\pi +
J/\psi$ dissociation cross section is predicted to be small, but the $\pi
+\psi'$ cross section is found to be quite large, in qualitative agreement with
experimental data on $J/\psi$ and $\psi'$ production in high-energy heavy-ion
collisions. The $\rho + J/\psi$ and $\rho + \psi'$ dissociation cross sections
are both predicted to be large numerically and are divergent near threshold.
These $\rho + J/\psi$ and $\rho + \psi'$ cross sections show considerable
sensitivity to the mass of the $\rho$ meson if it is off energy shell.
