  The formalism of Baier-Dokshitzer-Mueller-Peign\'e-Schiff and Zakharov
determines the modifications of parton splittings in the QCD plasma that arise
from medium-induced gluon radiation. Here, we study medium-modifications of the
gluon splitting into a quark--anti-quark pair in this BDMPS-Z formalism. We
derive a compact path-integral formulation that resums effects from an
arbitrary number of interactions with the medium to leading order in the
$1/N_c^2$ expansion. Analyses in the $N=1$ opacity and the saddle point
approximations reveal two phenomena: a medium-induced momentum broadening of
the relative quark--anti-quark pair momentum that increases the invariant mass
of quark--anti-quark pairs, and a medium-enhanced production of such pairs. We
note that both effects are numerically sizeable if the average momentum
transfer from the medium is comparable to the quark mass. In ultra-relativistic
heavy-ion collisions, this condition is satisfied for charm quarks. We
therefore focus our numerical analysis on the medium modification of $g\to
c\bar{c}$, although our derivation applies equally well to $g\to b\bar{b}$ and
to gluons splitting into light-flavoured quark--anti-quark pairs.
