Out of several dozen z > 7 candidate galaxies observed spectroscopically, only five have been confirmed via Lymanα emission, at z=7.008, 7.045, 7.109, 7.213 and 7.215. [1][2][3][4] The small fraction of confirmed galaxies may indicate that the neutral fraction in the intergalactic medium (IGM) rises quickly at z > 6.5, as Lymanα is resonantly scattered by neutral gas.3, 5-8 However, the small samples and limited depth of previous observations makes these conclusions tentative. Here we report the results of a deep near-infrared spectroscopic survey of 43 z > 6.5 galaxies. We detect only a single galaxy, confirming that some process is making Lymanα difficult to detect. The detected emission line at 1.0343 µm is likely to be Lymanα emission, placing this galaxy at a redshift z = 7.51, an epoch 700 million years after the Big Bang. This galaxy's colors are consistent with significant metal content, implying that galaxies become enriched rapidly. We measure a surprisingly high star formation rate of 330 M⊙ yr −1 , more than a factor of 100 greater than seen in the Milky Way. Such a galaxy is unexpected in a survey of our size 9 , suggesting that the early universe may harbor more intense sites of star-formation than expected.We obtained near-infrared (near-IR) spectroscopy of galaxies originally discovered in the Cosmic Assembly Near-infrared Deep Extragalactic Legacy Survey (CANDELS) 10, 11 with the newlycommissioned near-infrared spectrograph MOSFIRE 12 on the Keck I 10 meter telescope. From a parent sample of over 100 galaxy candidates at z > 7 selected via their HST colors through the photometric redshift technique [13][14][15][16] , we observed 43 candidate high-redshift galaxies over two MOSFIRE pointings with exposure times of 5.6 and 4.5 hr, respectively. Our observations covered Lyα emission at redshifts of 7.0 -8.2. We visually inspected the reduced data at the expected slit positions for our 43 observed sources and found plausible emission lines in eight objects, with only one line detected at >5σ significance. The detected emission line is at a wavelength of 1.0343µm with an integrated signal-to-noise (S/N) of 7.8 ( Figure 1) and comes from the object designated z8 GND 5296 in our sample (RA=12:36:37.90; Dec=62:18:08.5 J2000). Based on the arguments outlined below (and discussed extensively in the supplementary material), we identify this line as the Lyα transition of hydrogen at a line-peak redshift of z = 7.5078 ± 0.0004; consistent with our photometric redshift 68% confidence range of 7.5 < z < 7.9 for z8 GND 5296. As expected for a galaxy at z = 7.51, z8 GND 5296 is completely undetected in the HST optical bands, including an extremely deep 0.8 µm image (Figure 2). The galaxy is bright in the HST near-IR bands, becoming brighter with increasing wavelength, implying that the Lyman break lies near 1 µm and that the galaxy has a moderately red rest-frame UV color. The galaxy is well-detected in both Spitzer/IRAC bands and is much brighter at 4.5 µm than at 3.6 µm. The strong break at observed 1 µm res...