  The large number of $B_c$ mesons observed by LHCb suggests a sizable cross
section for producing doubly-heavy baryons in the same experiment. Motivated by
this, we estimate masses of the doubly-heavy $J=1/2$ baryons $\Xi_{cc}$,
$\Xi_{bb}$, and $\Xi_{bc}$, and their $J=3/2$ hyperfine partners, using a
method which accurately predicts the masses of ground-state baryons with a
single heavy quark. We obtain $M(\Xi_{cc}) = 3627 \pm 12$ MeV, $M(\Xi_{cc}^*)=
3690 \pm 12$ MeV, $M(\Xi_{bb}) = 10162 \pm 12$ MeV, $M(\Xi_{bb}^*)= 10184 \pm
12$ MeV, $M(\Xi_{bc}) = 6914 \pm 13$ MeV, $M(\Xi'_{bc}) = 6933 \pm 12$ MeV, and
$M(\Xi_{bc}^*) = 6969 \pm 14$ MeV. As a byproduct, we estimate the hyperfine
splitting between $B_c^*$ and $B_c$ mesons to be $68 \pm 8$ MeV. We discuss
P-wave excitations, production mechanisms, decay modes, lifetimes, and
prospects for detection of the doubly heavy baryons.
