A search for the dimuon decay of the Standard Model Higgs boson is performed using data corresponding to an integrated luminosity of 139 fb$^{-1}$ collected in Run 2 with the ATLAS detector in $pp$ collisions at $\sqrt{s}=13$ TeV at the Large Hadron Collider. No significant excess is observed above the expected background. The observed upper limit on the cross section times branching ratio is $1.7$ times the Standard Model prediction at 95% confidence level for a Higgs boson mass of 125.09 GeV, while the expected limit in the absence (assuming SM signal yield) of a $H\to\mu\mu$ signal is $1.3$ ($2.2$). The measured signal strength, defined as the ratio of the observed signal yield to the one expected in the Standard Model, is $\mu = 0.5 \pm 0.7$.

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