Collider searches for dark sectors, new particles interacting only feebly with ordinary matter, have largely focused on identifying signatures of new mediators, leaving much of dark sector structures unexplored. In particular, the existence of dark matter bound states (darkonia) remains to be investigated. This possibility could arise in a simple model in which a dark photon (A^{'}) is light enough to generate an attractive force between dark fermions. We report herein a search for a J^{PC}=1^{--} darkonium state, the ϒ_{D}, produced in the reaction e^{+}e^{-}→γϒ_{D}, ϒ_{D}→A^{'}A^{'}A^{'}, where the dark photons subsequently decay into pairs of leptons or pions, using 514 fb^{-1} of data collected with the BABAR detector. No significant signal is observed, and we set bounds on the γ-A^{'} kinetic mixing as a function of the dark sector coupling constant for 0.001<m_{A^{'}}<3.16 GeV and 0.05<m_{ϒ_{D}}<9.5 GeV.