We report the first observation of e+e-→ρ+ρ-, in a data sample of 379 fb-1 collected with the BABAR detector at the PEP-II e+e- storage ring at center-of-mass energies near √s=10.58 GeV. We measure a cross section of σ(e+e-→ρ+ρ-)=8.3±0.7(stat)±0.8(syst) fb within the range of |cosθ*|<0.8 and |cosθ±|<0.85, where θ* is the center-of-mass polar angle of the ρ± meson and θ± are the angles in the ρ± rest frame between the direction of the boost from the laboratory frame and the direction of the π±. Assuming production through single-photon annihilation, there are three independent helicity amplitudes. We measure the ratios of their squared moduli to be |F00|2∶|F10|2∶|F11|2= 0.51±0.14(stat)±0.07(syst)∶0.10±0.04(stat)±0.01(syst)∶0.04±0.03(stat)±0.01(syst). The |F00|2 result is inconsistent with the prediction of 1.0 made by QCD models with a significance of 3.1 standard deviations including systematic uncertainties.