We present new upper limits on the volume-weighted neutral hydrogen fraction, $\bar{x}_{\rm H\,\small {i}}$ , at z ∼ 5–6 derived from spectroscopy of bright quasars. The fraction of the Lyα and Lyβ forests that is ‘dark’ (with zero flux) provides the only model-independent upper limit on $\bar{x}_{\rm H\,\small {i}}$ , 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 $\bar{x}_{\rm H\,\small {i}}$ derived from dark pixel analysis to $\bar{x}_{\rm H\,\small {i}}\le 0.06{+0.05 (1\sigma )}$ at z = 5.9 and $\bar{x}_{\rm H\,\small {i}}\le 0.04{+0.05 (1\sigma )}$ 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.