  The production of a $D\bar D$ meson-pair in antiproton-proton ($\bar p p$)
annihilation close to the production threshold is investigated, with special
emphasis on the role played by the $\psi$(3770) resonance. The study is
performed in a meson-baryon model where the elementary charm production process
is described by baryon exchange. Effects of the interactions in the initial and
final states are taken into account rigorously, where the latter involves also
those due to the $\psi$(3770). The predictions for the $D\bar D$ production
cross section are in the range of 30 -- 250 nb, the contribution from the
$\psi$(3770) resonance itself amounts to roughly 20 -- 80 nb.
