  Lyman-alpha (Lya) emission of neutral hydrogen (1215.67 Angstr\"om) is the
main contributor to the ultraviolet flux of low-mass stars such as M dwarfs. It
is also the main light source used in studies of the evaporating upper
atmospheres of transiting extrasolar planets with ultraviolet transmission
spectroscopy. However, there are very few observations of the Lya emissions of
quiet M dwarfs, and none exist for those hosting exoplanets. Here, we present
Lya observations of the hot-neptune host star GJ436 with the Hubble Space
Telescope Imaging Spectrograph (HST/STIS). We detect bright emission in the
first resolved and high quality spectrum of a quiet M dwarf at Lya. Using an
energy diagram for exoplanets and an N-body particle simulation, this detection
enables the possible exospheric signature of the hot neptune to be estimated as
a ~11% absorption in the Lya stellar emission, for a typical mass-loss rate of
10^10 g/s. The atmosphere of the planet GJ436b is found to be stable to
evaporation, and should be readily observable with HST. We also derive a
correlation between X-ray and Lya emissions for M dwarfs. This correlation will
be useful for predicting the evaporation signatures of planets transiting other
quiet M dwarfs.
