  We compute the masses of the singly and doubly charmed baryons in full QCD
using the relativistic Fermilab action for the charm quark. For the light
quarks we use domain-wall fermions in the valence sector and improved
Kogut-Susskind sea quarks. We use the low-lying charmonium spectrum to tune our
heavy-quark action and as a guide to understanding the discretization errors
associated with the heavy quark. Our results are in good agreement with
experiment within our systematics. For the \Xi_{cc}, we find the
isospin-averaged mass to be M_{\Xi_{cc}} = 3665 +/- 17 +/- 14 +0 -78 MeV; the
three given uncertainties are statistical, systematic and an estimate of
lattice discretization errors, respectively. In addition, we predict the mass
splitting of the (isospin-averaged) spin-1/2 \Omega_{cc} with the \Xi_{cc} to
be M_{\Omega_{cc}} - M_{\Xi_{cc}} = 98 +/- 9 +/- 22 +/- 13 MeV (in this mass
splitting, the leading discretization errors are also suppressed by SU(3)
symmetry). Combining this splitting with our determination of M_{\Xi_{cc}}
leads to our prediction of the spin-1/2 \Omega_{cc} mass, M_{\Omega_{cc}} =
3763 +/- 19 +/- 26 +13 -79 MeV.
