We report the direct measurement of the ^7Be solar neutrino signal rate performed with the Borexino detector at the Laboratori Nazionali del Gran Sasso. The interaction rate of the 0.862 MeV ^7Be neutrinos is 49+-3(stat)+-4(syst) counts/(day * 100ton). The hypothesis of no oscillation for ^7Be solar neutrinos is inconsistent with our measurement at the 4sigma level. Our result is the first direct measurement of the survival probability for solar nu_e in the transition region between matter-enhanced and vacuum-driven oscillations. The measurement improves the experimental determination of the flux of ^7Be, pp, and CNO solar nu_e, and the limit on the magnetic moment of neutrinos.