A search for a dark photon, a new light neutral particle, which decays promptly into collimated pairs of electrons or muons is presented. The search targets dark photons resulting from the exotic decay of the Standard Model Higgs boson, assuming its production via the dominant gluon-gluon fusion mode. The analysis is based on $140~\hbox {fb}^{-1}$ of data collected with the ATLAS detector at the Large Hadron Collider from proton-proton collisions at a center-of-mass energy of 13 $\text {T}\text {e}\hspace{-1.00006pt}\text {V}$. Events with collimated pairs of electrons or muons are analysed and background contributions are estimated using data-driven techniques. No significant excess in the data above the Standard Model background is observed. Upper limits are set at 95% confidence level on the branching ratio of the Higgs boson decay into dark photons between 0.001% and 5%, depending on the assumed dark photon mass and signal model.