A search for events with a dark photon produced in association with a dark Higgs boson via rare decays of the standard model Z boson is presented, using 139 fb^{-1} of sqrt[s]=13 TeV proton-proton collision data recorded by the ATLAS detector at the Large Hadron Collider. The dark boson decays into a pair of dark photons, and at least two of the three dark photons must each decay into a pair of electrons or muons, resulting in at least two same-flavor opposite-charge lepton pairs in the …