  Using a sample of $(1310.6\pm7.0)\times10^6$ $J/\psi$ events recorded with
the BESIII detector at the symmetric electron positron collider BEPCII, we
report the observation of the decay of the $(1^1 S_0)$ charmonium state
$\eta_c$ into a pair of $\omega$ mesons in the process
$J/\psi\to\gamma\omega\omega$. The branching fraction is measured for the first
time to be $\mathcal{B}(\eta_c\to\omega\omega)=
(2.88\pm0.10\pm0.46\pm0.68)\times10^{-3}$, where the first uncertainty is
statistical, the second systematic and the third is from the uncertainty of
$\mathcal{B}(J/\psi\to\gamma\eta_c)$. The mass and width of the $\eta_c$ are
determined as $M=(2985.9\pm0.7\pm2.1)\,$MeV/$c^2$ and
$\Gamma=(33.8\pm1.6\pm4.1)\,$MeV.
