  Production and decay of $\Omega_c^0$ baryons is studied with $\sim230
\mathrm{fb}^{-1}$ of data recorded with the BABAR detector at the PEP-II
$e^+e^-$ asymmetric-energy storage ring at SLAC. The $\Omega_c^0$ is
reconstructed through its decays into $\Omega^-\pi^+$,
$\Omega^-\pi^+\pi^+\pi^-$, and $\Xi^-K^-\pi^+\pi^+$ final states.
  The invariant mass spectra are presented and the signal yields are extracted.
Ratios of branching fractions are measured relative to the $\Omega^-\pi^+$ mode
\frac{{\cal B}(\Xi^-K^-\pi^+\pi^+)}{{\cal B}(\Omega^-\pi^+)}
  &= 0.31 \pm 0.15\mathrm{(stat.)}
  \pm 0.04\mathrm{(syst.)},
  \frac{{\cal B}(\Omega^-\pi^+\pi^+\pi^-)}{{\cal B}(\Omega^-\pi^+)}
  &< 0.30\qquad (90%\mathrm{CL}).
  The momentum spectrum (not corrected for efficiency) of $\Omega_c^0$ baryons
is extracted from decays into $\Omega^-\pi^+$, establishing the first
observation of $\Omega_c^0$ production from B decays.
