We present the results of a search for charged-lepton-flavor violating decays
$\tau^{-} \rightarrow \ell^{-}K_{S}^{0}$, where $\ell^{-}$ is either an
electron or a muon. We combine $e^+e^-$ data samples recorded by the Belle II
experiment at the SuperKEKB collider (428 fb$^{-1}$) with samples recorded by
the Belle experiment at the KEKB collider (980 fb$^{-1}$) to obtain a sample of
1.3 billion $e^+e^-\to\tau^+\tau^-$ events. We observe 0 Und 1 events and set
$90\%$ confidence level upper limits of $0.8 \times 10^{-8}$ Und $1.2 \times
10^{-8}$ on the branching fractions of the decay modes $\tau^{-} \rightarrow
e^{-}K_{S}^{0}$ and $\tau^{-} \rightarrow \mu^{-}K_{S}^{0}$, respectively.
These are the most stringent upper limits to date.

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