  We report a measurement of the lifetime of the $\Omega_c^0$ baryon using
proton-proton collision data at center-of-mass energies of 7 and 8~TeV,
corresponding to an integrated luminosity of 3.0 fb$^{-1}$ collected by the
LHCb experiment. The sample consists of about 1000
$\Omega_b^-\to\Omega_c^0\mu^-\bar{\nu}_{\mu} X$ signal decays, where the
$\Omega_c^0$ baryon is detected in the $pK^-K^-\pi^+$ final state and $X$
represents possible additional undetected particles in the decay. The
$\Omega_c^0$ lifetime is measured to be $\tau_{\Omega_c^0} = 268\pm24\pm10\pm2$
fs, where the uncertainties are statistical, systematic, and from the
uncertainty in the $D^+$ lifetime, respectively. This value is nearly four
times larger than, and inconsistent with, the current world-average value.
