  We present 36 spectroscopically confirmed intrinsically UV-faint Ly$\alpha$
emitting galaxies from follow-up observations with Keck/DEIMOS of
gravitationally lensed high-redshift candidates. Candidates were selected to be
between $5\lesssim z \lesssim 7$ from photometric data using \textit{HST} and
\textit{Spitzer} imaging surveys. We used photometric redshift information to
perform an integrated photometric redshift probability cut $>1\%$ between
$5<z<7$ to construct a sample of 198 high-redshift objects. Our high-redshift
sample spans intrinsic UV luminosities from a few $L^*$ down to $0.001L^*$. We
identified 19 high-confidence detections of Ly$\alpha$ and an additional 17
likely detections. We divided our sample into lower-redshift ($z\sim5.5$) and
higher-redshift ($z\sim6.5$) bins and ran Monte Carlo trials, incorporating the
strength of the Ly$\alpha$ emission and the photometric redshift of the
non-detections. Considering only objects where Ly$\alpha$ could be detected at
EW(Ly$\alpha$)$>$25{\AA} at $3\sigma$ at the fiducial depth of our survey, and
only those galaxies with EW(Ly$\alpha$)$>$25{\AA} as true LAEs, and finally,
only objects with $m_{AB}<26.8$, we found the LAE fraction to be flat, or
modestly increase from 0.26$\pm0.04$ to 0.30$\pm0.04$. These values relative to
those for lower-redshift samples are consistent with a rising LAE fraction with
redshift out to $z\sim6$, but at $z\sim6.5$ there is some tension between our
results and results from surveys at intrinsically brighter luminosities. We
conclude intrinsically fainter galaxies have Ly$\alpha$ emission, and there is
a steep drop in the LAE fraction from our high-redshift sample at $z\sim6.5$
and from similar galaxies at $z\sim7.5$. This likely indicates we are
witnessing the tail end of the epoch of reionization, as such a drop is not
expected due to changes of intrinsic galaxy properties between these redshifts.
