  To date, more than 750 planets have been discovered orbiting stars other than
the Sun. Two sub-classes of these exoplanets, "hot Jupiters" and their less
massive counterparts "hot Neptunes," provide a unique opportunity to study the
extended atmospheres of planets outside of our solar system. We describe here
the first far-ultraviolet transit study of a hot Neptune, specifically GJ436b,
for which we use HST/STIS Lyman-$\alpha$ spectra to measure stellar flux as a
function of time, observing variations due to absorption from the planetary
atmosphere during transit. This analysis permits us to derive information about
atmospheric extent, mass-loss rate from the planet, and interactions between
the star and planet. We observe an evolution of the Lyman-$\alpha$ lightcurve
with a transit depth of GJ436b from $8.8\pm4.5\%$ near mid-transit, to
$22.9\pm3.9\%$ $\sim2$ hours after the nominal geometric egress of the planet.
Using data from the time-tag mode and considering astrophysical noise from
stellar variability, we calculate a post-egress occultation of $23.7\pm4.5\%$,
demonstrating that the signature is statistically significant and of greater
amplitude than can be attributed to stellar fluctuations alone. The extended
egress absorption indicates the probable existence of a comet-like tail
trailing the exoplanet. We calculate a mass-loss rate for GJ436b in the range
of $3.7\times10^6 -1.1\times10^{9}$ g s$^{-1}$, corresponding to an atmospheric
lifetime of $4\times10^{11}-2\times10^{14}$ years.
