  The KamLAND-Zen experiment has provided stringent constraints on the
neutrinoless double-beta ($0\nu\beta\beta$) decay half-life in $^{136}$Xe using
a xenon-loaded liquid scintillator. We report an improved search using an
upgraded detector with almost double the amount of xenon and an ultralow
radioactivity container, corresponding to an exposure of 970 kg yr of
$^{136}$Xe. These new data provide valuable insight into backgrounds,
especially from cosmic muon spallation of xenon, and have required the use of
novel background rejection techniques. We obtain a lower limit for the
$0\nu\beta\beta$ decay half-life of $T_{1/2}^{0\nu} > 2.3 \times 10^{26}$ yr at
90% C.L., corresponding to upper limits on the effective Majorana neutrino mass
of 36-156 meV using commonly adopted nuclear matrix element calculations.
