If we assume that there is no anomalously large $\mathrm{CP}$-violating $K_S^0\to {\mu }^{+}{\mu }^{-}$ decay mode, then a possible conflict with a unitarity bound could arise similar to the present situation with $K_L^0\to {\mu }^{+}{\mu }^{-}$. It is therefore important to have a reliable estimate of the electromagnetic contribution for $K_S^0\to {\mu }^{+}{\mu }^{-}$. In this paper, we assume $CP$ invariance (so $K_S^0\equiv K_1^0$) and give the results of a detailed calculation of both the real and imaginary parts of the $K_1^0\to {\mu }^{+}{\mu }^{-}$ amplitude based on contributions from $2\gamma$, ${\pi }^{+}{\pi }^{-}$, and ${\pi }^{+}{\pi }^{-}\gamma$ states.

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