  We report the first measurement of the charm-mixing parameter $y_{CP}$ in
$D^{0}$ decays to the $CP$-odd final state $K^{0}_{S}\omega$. The study uses
the full Belle $e^{+}e^-$ annihilation data sample of $976\rm~ fb^{-1}$ taken
at or near the $\Upsilon(4S)$ centre-of-mass energy. We find $y_{CP} = (0.96
\pm 0.91 \pm 0.61 {}^{+0.17}_{-0.00})\%$, where the first uncertainty is
statistical, the second is systematic due to event selection and background,
and the last is due to possible presence of $CP$-even decays in the data
sample.
