The process e+e−→π+π−π0γ is studied at a center-of-mass energy near the ϒ(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−→π+π−π0 cross section in the center-of-mass energy range from 0.62 to 3.5 GeV. In the energy regions of the ω and ϕ 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−→π+π−π0 cross section from threshold to 2.0 GeV is (45.86±0.14±0.58)×10−10. From the fit to the measured 3π mass spectrum we have determined the resonance parameters Γ(ω→e+e−)B(ω→π+π−π0)=(0.5698±0.0031±0.0082) keV, Γ(ϕ→e+e−)B(ϕ→π+π−π0)=(0.1841±0.0021±0.0080) keV, and B(ρ→3π)=(0.88±0.23±0.30)×10−4. The significance of the ρ→3π signal is greater than 6σ. For the J/ψ resonance we have measured the product Γ(J/ψ→e+e−)B(J/ψ→3π)=(0.1248±0.0019±0.0026) keV.21 MoreReceived 4 October 2021Accepted 12 November 2021DOI:https://doi.org/10.1103/PhysRevD.104.112003Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP3.Published by the American Physical SocietyPhysics Subject Headings (PhySH)Research AreasParticle interactionsPhysical SystemsLight mesonsQuarkoniaTechniquesLepton collidersParticle productionParticles & Fields