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