We investigated the atomic configuration of the oxygen negative-$U$ center in GaAs, which has a Ga-O-Ga structure, based on first-principles calculations. We calculated the stable atomic configurations and formation energies for two Ga-O-Ga structures. One is $\mathrm{Ga}\ensuremath{-}{\mathrm{O}{\ensuremath{-}V}_{\mathrm{As}}}\ensuremath{-}\mathrm{Ga},$ which has been proposed as the atomic structure of the negative-$U$ center. The other is ${\mathrm{G}\mathrm{a}\ensuremath{-}\mathrm{O}}_{i}\ensuremath{-}\mathrm{Ga},$ which we propose in this paper. Five charge states from $+1$ to $\ensuremath{-}3$ were considered. For both $\mathrm{Ga}\ensuremath{-}{\mathrm{O}{\ensuremath{-}V}_{\mathrm{As}}}\ensuremath{-}\mathrm{Ga}$ and ${\mathrm{G}\mathrm{a}\ensuremath{-}\mathrm{O}}_{i}\ensuremath{-}\mathrm{Ga},$ the stable atomic configurations depend on the charge states. For $\mathrm{Ga}\ensuremath{-}{\mathrm{O}{\ensuremath{-}V}_{\mathrm{As}}}\ensuremath{-}\mathrm{Ga},$ the calculated atomic configurations cannot explain the experimentally obtained tendency in the local-vibrational-mode absorption frequencies for different charge states. The calculated formation energy as a function of the Fermi level indicates that the $\mathrm{Ga}\ensuremath{-}{\mathrm{O}{\ensuremath{-}V}_{\mathrm{As}}}\ensuremath{-}\mathrm{Ga}$ structure does not have a negative-$U$ nature. In contrast, the calculated results of the ${\mathrm{G}\mathrm{a}\ensuremath{-}\mathrm{O}}_{i}\ensuremath{-}\mathrm{Ga}$ structure well explain the experimental results such as the local-vibrational-mode absorption frequencies for different charge states and the negative-$U$ nature. Thus, it is concluded that the atomic configuration of the negative-$U$ center in GaAs is ${\mathrm{G}\mathrm{a}\ensuremath{-}\mathrm{O}}_{i}\ensuremath{-}\mathrm{Ga},$ not $\mathrm{Ga}\ensuremath{-}{\mathrm{O}{\ensuremath{-}V}_{\mathrm{As}}}\ensuremath{-}\mathrm{Ga}.$