  The $J/\psi \pi\to \bar{D} D^*, \bar{D} D$, ${\bar D}^* D^*$ and ${\bar D}
D^*$ cross sections as a function of $\sqrt{s}$ are evaluated in a QCD sum rule
calculation. We study the Borel sum rule for the four point function involving
pseudoscalar and vector meson currents, up to dimension four in the operator
product expansion. We find that our results are smaller than the $J/\psi \pi\to
{charmed mesons}$ cross sections obtained with models based on meson exchange,
but are close to those obtained with quark exchange models.
