  The production of single-charmed baryons $\Lambda_c^+ \Lambda_c^-$,
$\Lambda_c^+ \Sigma_c^-$+h.c. and $\Sigma_c^+ \Sigma_c^-$ in $p\bar p$
collisions is studied in the framework of a constituent quark model which has
satisfactorily described the $N\bar N$ system and the strangeness production
$p\bar p\to \Lambda\bar \Lambda$, $\Lambda\bar\Sigma$ and $\Sigma\bar\Sigma$
processes. Predictions on the total cross sections are analyzed for different
approaches to the underlying $n\bar n\to c\bar c$ process, mediated by one
gluon annihilation diagrams. The results indicate that the cross section is of
the order of $1$ nb between $10-14$ GeV for $\Lambda_c^+\Lambda_c^-$ and
$\Lambda_c^+ \Sigma_c^-$ channels, and around $0.01-0.1$ nb for
$\Sigma_c^+\Sigma_c^-$. This estimations can be relevant for their future
search in facilities like $\bar{P}$ANDA.
