We study the process e^+e^-→π^+π^-π^+π^-γ, with a photon emitted from the initial-state electron or positron, using 454.3 fb^(-1) of data collected with the BABAR detector at SLAC, corresponding to approximately 260 000 signal events. We use these data to extract the nonradiative σ(e^+e^-→π^+π^-π^+π^-) cross section in the energy range from 0.6 to 4.5 GeV. The total uncertainty of the cross section measurement in the peak region is less than 3%, higher in precision than the corresponding results obtained from energy scan data
