  We present a large spectroscopic campaign with Keck/MOSFIRE targeting
Lyman-alpha emission (Ly$\alpha$) from intrinsically faint Lyman-break Galaxies
(LBGs) behind 12 efficient galaxy cluster lenses. Gravitational lensing allows
us to probe the more abundant faint galaxy population to sensitive Ly$\alpha$
equivalent width limits. During the campaign we targeted 70 LBG candidates with
MOSFIRE Y-band, selected photometrically to cover Ly$\alpha$ over the range
$7<z<8.2$. We detect $S/N>5$ emission lines in 2 of these galaxies and find
that they are likely Ly$\alpha$ at $z=7.148\pm0.001$ and $z=7.161\pm0.001$. We
present new lens models for 4 of the galaxy clusters, using our previously
published lens models for the remaining clusters to determine the magnification
factors for the source galaxies. Using a Bayesian framework that employs large
scale reionization simulations of the intergalactic medium (IGM) as well as
realistic properties of the interstellar medium and circumgalactic medium, we
infer the volume-averaged neutral hydrogen fraction,
$\overline{x}_{\mathrm{HI}}$, in the IGM during reionization to be
$\overline{x}_{\mathrm{HI}}=0.88^{+0.05}_{-0.10}$ at $z=7.6\pm0.6$. Our result
is consistent with a late and rapid reionization scenario inferred by Planck.
