  We present an improved search for neutrinoless double-beta ($0\nu\beta\beta$)
decay of $^{136}$Xe in the KamLAND-Zen experiment. Owing to purification of the
xenon-loaded liquid scintillator, we achieved a significant reduction of the
$^{110m}$Ag contaminant identified in previous searches. Combining the results
from the first and second phase, we obtain a lower limit for the
$0\nu\beta\beta$ decay half-life of $T_{1/2}^{0\nu} > 1.07 \times 10^{26}$ 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
quasi-degenerate neutrino mass region.
