We present an improved search for neutrinoless double-beta (0νββ) decay of Xe136 in the KamLAND-Zen experiment. Owing to purification of the xenon-loaded liquid scintillator, we achieved a significant reduction of the Ag110m contaminant identified in previous searches. Combining the results from the first and second phase, we obtain a lower limit for the 0νββ decay half-life of T1/20ν>1.07×1026 yr at 90% C.L., an almost sixfold improvement over previous limits. Using commonly adopted nuclear matrix element calculations, the corresponding upper limits on the effective Majorana neutrino mass are in the range 61-165 meV. For the most optimistic nuclear matrix elements, this limit reaches the bottom of the quasidegenerate neutrino mass region.