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.