Searches for the decay of the Higgs boson $H \to ee$ and $H \to e \mu$ are performed using data corresponding to an integrated luminosity of 139 bf$^{-1}$ collected with the ATLAS detector in $pp$ collisions at $\sqrt{s}=$13 TeV at the LHC. No significant signals are observed, in agreement with the Standard Model expectation. For a Higgs boson mass of 125 GeV the observed (expected) upper limit at the 95% confidence level on the branching fraction of $H \to ee$ is $3.6 \times 10^{-4}$ ($3.5 \times 10^{-4}$) and on $H \to e\mu$ is $6.1\times 10^{-5}$ ($5.8\times 10^{-5}$). These results represent an improvement of a factor of about 5 and 6 on the previous best limits on the branching fractions of $H \to ee$ and $H \to e \mu$, respectively.

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