Within one billion years of the Big Bang, intergalactic hydrogen was ionized by sources emitting ultraviolet and higher energy photons. This was the final phenomenon to globally affect all the baryons (visible matter) in the Universe. It is referred to as cosmic reionization and is an integral component of cosmology. It is broadly expected that intrinsically faint galaxies were the primary ionizing sources due to their abundance in this epoch. However, at the highest redshifts ($z>7.5$; lookback time 13.1 Gyr), all galaxies with spectroscopic confirmations to date are intrinsically bright and, therefore, not necessarily representative of the general population. Here, we report the unequivocal spectroscopic detection of a low luminosity galaxy at $z>7.5$. We detected the Lyman-$alpha$ emission line at $sim 10504$ {AA} in two separate observations with MOSFIRE on the Keck I Telescope and independently with the Hubble Space Telescopes slit-less grism spectrograph, implying a source redshift of $z = 7.640 pm 0.001$. The galaxy is gravitationally magnified by the massive galaxy cluster MACS J1423.8+2404 ($z = 0.545$), with an estimated intrinsic luminosity of $M_{AB} = -19.6 pm 0.2$ mag and a stellar mass of $M_{star} = 3.0^{+1.5}_{-0.8} times 10^8$ solar masses. Both are an order of magnitude lower than the four other Lyman-$alpha$ emitters currently known at $z > 7.5$, making it probably the most distant representative source of reionization found to date.
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