  Extreme Scattering Events (ESEs) are attributed to radio-wave refraction by a
cloud of free-electrons crossing the line-of-sight. We present a new model in
which these electrons form the photo-ionized 'skin' of an underlying cool,
self-gravitating cloud in the Galactic halo. In this way we avoid the severe
over-pressure problem which afflicts other models. The UV flux in the Galactic
halo naturally generates electron densities of the right order. We demonstrate,
for the first time, a good reproduction of the prototypical ESE in the quasar
0954+658. The neutral clouds are a few AU in radius and have masses less than
about 10^{-3} solar. The observed rate of ESEs implies that a large fraction of
the mass of the Galaxy is in this form.
