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±3stat±4syst counts/(day⋅100 ton). The hypothesis of no oscillation for 7Be solar neutrinos is inconsistent with our measurement at the 4σ C.L. Our result is the first direct measurement of the survival probability for solar νe 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 νe, and the limit on the effective neutrino magnetic moment using solar neutrinos.