  We present a measurement of time-dependent rate asymmetries in $B^0\to \phi
K^0_S$ decays to search for non-standard-model physics in $b\to q
\overline{q}s$ transitions. The data sample is collected with the Belle II
detector at the SuperKEKB asymmetric-energy $e^{+}e^{-}$ collider in 2019-2022
and contains $(387\pm 6)\times 10^6$ bottom-antibottom mesons from
$\Upsilon(4S)$ resonance decays. We reconstruct $162\pm17$ signal events and
extract the charge-parity ($CP$) violating parameters from a fit to the
distribution of the proper-decay-time difference of the two $B$ mesons. The
measured direct and mixing-induced $CP$ asymmetries are $A=0.31\pm0.20\pm0.05$
and $S=0.54\pm0.26^{+0.06}_{-0.08}$, respectively, where the first
uncertainties are statistical and the second are systematic. The results are
compatible with the $CP$ asymmetries observed in $b\to c\overline{c} s$
transitions.
