We report a measurement of the neutrino-electron elastic scattering rate of 862 keV 7Be solar neutrinos based on a 165.4 kton-day exposure of KamLAND. The observed rate is 582 +/- 90 (kton-day)^-1, which corresponds to a 862 keV 7Be solar neutrino flux of (3.26 +/- 0.50) x 10^9 cm^-2s^-1, assuming a pure electron flavor flux. Comparing this flux with the standard solar model prediction and further assuming three flavor mixing, a nu_e survival probability of 0.66 +/- 0.14 is determined from the KamLAND data. Utilizing a global three flavor oscillation analysis, we obtain a total 7Be solar neutrino flux of (5.82 +/- 0.98) x 10^9 cm^-2s^-1, which is consistent with the standard solar model predictions.