Cosmic reionization was the last major phase transition of hydrogen from neutral to highly ionized in the intergalactic medium (IGM). Current observations show that the IGM is significantly neutral atz> 7 and largely ionized byz∼ 5.5. However, most methods to measure the IGM neutral fraction are highly model dependent and are limited to when the volume-averaged neutral fraction of the IGM is either relatively low ($x¯HI≲10−3$) or close to unity ($x¯HI∼1$). In particular, the neutral fraction evolution of the IGM at the critical redshift range ofz= 6–7 is poorly constrained. We present new constraints on$x¯HI$atz∼ 5.1–6.8 by analyzing deep optical spectra of 53 quasars at 5.73 <z< 7.09. We derive model-independent upper limits on the neutral hydrogen fraction based on the fraction of “dark” pixels identified in the Lyαand Lyβforests, without any assumptions on the IGM model or the intrinsic shape of the quasar continuum. They are the first model-independent constraints on the IGM neutral hydrogen fraction atz∼ 6.2–6.8 using quasar absorption measurements. Our results give upper limits of