The process e(+)e(-) -> pi(+) pi(-) pi(0)gamma is studied at a center-of-mass energy near the Upsilon(4S) resonance using a data sample of 469 fb(-1) collected with the BABAR detector at the PEP-II collider. We have performed a precise measurement of the e(+)e(-) -> pi(+) pi(-) pi(0) cross section in the center-of-mass energy range from 0.62 to 3.5 GeV. In the energy regions of the omega and phi resonances, the cross section is measured with a systematic uncertainty of 1.3%. The leading-order hadronic contribution to the muon magnetic anomaly calculated using the measured e(+) e(-) -> pi(+) pi(-) pi(0) cross section from threshold to 2.0 GeV is (45.86 +/- 0.14 +/- 0.58) x 10(-10). From the fit to the measured 3 pi mass spectrum we have determined the resonance parameters Gamma(omega -> e(+)e(-)) B(omega -> pi(+) pi- pi(0)) = (0.5698 +/- 0.0031 +/- 0.0082) keV, Gamma(phi -> e(+)e(-)) B(phi -> pi(+) pi(-)pi(0)) = (0.1841 +/- 0.0021 +/- 0.0080) keV, and B(rho -> 3 pi) = (0.88 +/- 0.23 +/- 0.30) x 10(-4). The significance of the rho -> 3 pi signal is greater than 6 sigma. For the J/psi resonance we have measured the product Gamma(J/psi -> e(+) e(-)) B (J/psi -> 3 pi) = (0.1248 +/- 0.0019 +/- 0.0026) keV.
