A precise measurement of the cross section of the process e⁺e⁻ → π⁺π⁻ (γ) from threshold to an energy of 3 GeV is obtained with the initial-state radiation (ISR) method using 232 fb⁻¹ of data collected with the BABAR detector at e⁺e⁻ center-of-mass energies near 10.6 GeV. The ISR luminosity is determined from a study of the leptonic process e⁺e⁻ → μ⁺μ⁻ (γ) γ_{ISR}, which is found to agree with the next-to-leading-order QED prediction to within 1.1%. The cross section for the process e⁺e⁻ → π⁺π⁻ (γ) is obtained with a systematic uncertainty of 0.5% in the dominant ρ resonance region. The leading-order hadronic contribution to the muon magnetic anomaly calculated using the measured π π cross section from threshold to 1.8 GeV is (514.1 ± 2.2(stat) ± 3.1(syst))×10⁻¹⁰.