© 2015 American Physical Society. We report a measurement of the neutrino-electron elastic scattering rate of 862 keV Be7 solar neutrinos based on a 165.4 kt d exposure of KamLAND. The observed rate is 582±94(kt d)-1, which corresponds to an 862-keV Be7 solar neutrino flux of (3.26±0.52)×109cm-2s-1, assuming a pure electron-flavor flux. Comparing this flux with the standard solar model prediction and further assuming three-flavor mixing, a νe survival probability of 0.66±0.15 is determined from the KamLAND data. Utilizing a global three-flavor oscillation analysis, we obtain a total Be7 solar neutrino flux of (5.82±1.02)×109cm-2s-1, which is consistent with the standard solar model predictions.