Optical and near-infrared (NIR) spectra are presented for ULAS J131911.29+095051.4 (hereafter ULAS J1319+0950), a new redshift z = 6.127 ± 0.004 quasar discovered in the Third Data Release (DR3) of the UKIRT Infrared Deep Sky Survey (UKIDSS). The source has $Y_{\rm{Vega}}$ = 19.10 ± 0.03, corresponding to $M_{1450,{\rm{AB}}}$ = -27.12, which is comparable to the absolute magnitudes of the z $\simeq$ 6 quasars discovered in the Sloan Digital Sky Survey (SDSS). ULAS J1319+0950 was, in fact, registered by SDSS as a faint source with $z_{\rm{AB}}$ = 20.13 ± 0.12, just below the signal-to-noise ratio limit of the SDSS high-redshift quasar survey. The faint z-band magnitude is a consequence of the weak Ly α/N v emission line, which has a rest-frame equivalent width of ~20Å and provides only a small boost to the z-band flux. Nevertheless, there is no evidence of a significant new population of high-redshift quasars with weak emission lines from this UKIDSS-based search. The Ly α optical depth to ULAS J1319+0950 is consistent with that measured towards similarly distant SDSS quasars, implying that results from optical- and NIR-selected quasars may be combined in studies of cosmological reionization. Also presented is a new NIR-spectrum of the previously discovered UKIDSS quasar ULAS J020332.38+001229.2, which reveals the object to be a broad absorption line quasar. The new spectrum shows that the emission line initially identified as Ly α is actually N v, leading to a revised redshift of z = 5.72, rather than z = 5.86 as previously estimated.