  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 Telescope's 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.
