  The newly detected Earth-mass planet in the habitable zone of Proxima
Centauri could potentially host life - if it has an atmosphere that supports
surface liquid water. We show that thermal phase curve observations with the
James Webb Space Telescope (JWST) from 5-12 microns can be used to test the
existence of such an atmosphere. We predict the thermal variation for a bare
rock versus a planet with 35% heat redistribution to the nightside and show
that a JWST phase curve measurement can distinguish between these cases at
$4\sigma$ confidence, assuming photon-limited precision. We also consider the
case of an Earth-like atmosphere, and find that the ozone 9.8 micron band could
be detected with longer integration times (a few months). We conclude that JWST
observations have the potential to put the first constraints on the possibility
of life around the nearest star to the Solar System.
