  We present new upper limits on the volume-weighted neutral hydrogen fraction,
<xHI>, at z~5-6 derived from spectroscopy of bright quasars. The fraction of
the Lyman-alpha and Lyman-beta forests that is "dark" (with zero flux) provides
the only model-independent upper limit on <xHI>, requiring no assumptions about
the physical conditions in the intergalactic medium or the quasar's unabsorbed
UV continuum. In this work we update our previous results using a larger sample
(22 objects) of medium-depth (~ few hours) spectra of high-redshift quasars
obtained with the Magellan, MMT, and VLT. This significantly improves the upper
bound on <xHI> derived from dark pixel analysis to <xHI> <= 0.06 + 0.05
(1{\sigma}) at z=5.9, and <xHI> <= 0.04 + 0.05 (1{\sigma}) at z=5.6. These
results provide robust constraints for theoretical models of reionization, and
provide the strongest available evidence that reionization has completed (or is
very nearly complete) by z~6.
