  We present the results of a calculation of the positive-parity ground-state
charmed-baryon spectrum using 2+1+1 flavors of dynamical quarks. The
calculation uses a relativistic heavy-quark action for the valence charm quark,
clover-Wilson fermions for the valence light and strange quarks, and HISQ sea
quarks. The spectrum is calculated with a lightest pion mass around 220 MeV,
and three lattice spacings (a \approx 0.12 fm, 0.09 fm, and 0.06 fm) are used
to extrapolate to the continuum. The light-quark mass extrapolation is
performed using heavy-hadron chiral perturbation theory up to O(m_pi^3) and at
next-to-leading order in the heavy-quark mass. For the well-measured charmed
baryons, our results show consistency with the experimental values. For the
controversial J=1/2 Xi_{cc}, we obtain the isospin-averaged value
M_{Xi_{cc}}=3595(39)(20)(6) MeV (the three uncertainties are statistics,
fitting-window systematic, and systematics from other lattice artifacts, such
as lattice scale setting and pion-mass determination), which shows a 1.7 sigma
deviation from the experimental value. We predict the yet-to-be-discovered
doubly and triply charmed baryons Xi_{cc}^*, Omega_{cc}, Omega_{cc}^* and
Omega_{ccc} to have masses 3648(42)(18)(7) MeV, 3679(40)(17)(5) MeV,
3765(43)(17)(5) MeV and 4761(52)(21)(6) MeV, respectively.
