  We present a new joint analysis of pulsar dispersion measures and diffuse
H-alpha emission in the Milky Way, which we use to derive the density, pressure
and filling factor of the thick disk component of the warm ionised medium (WIM)
as a function of height above the Galactic disk. By excluding sightlines at low
Galactic latitude that are contaminated by HII regions and spiral arms, we find
that the exponential scale-height of free electrons in the diffuse WIM is 1830
(+120, -250) pc, a factor of two larger than has been derived in previous
studies. The corresponding inconsistent scale heights for dispersion measure
and emission measure imply that the vertical profiles of mass and pressure in
the WIM are decoupled, and that the filling factor of WIM clouds is a geometric
response to the competing environmental influences of thermal and non-thermal
processes. Extrapolating the properties of the thick-disk WIM to mid-plane, we
infer a volume-averaged electron density 0.014 +- 0.001 cm^-3, produced by
clouds of typical electron density 0.34 +- 0.06 cm^-3 with a volume filling
factor 0.04 +- 0.01. As one moves off the plane, the filling factor increases
to a maximum of ~30% at a height of approximately 1-1.5 kpc, before then
declining to accommodate the increasing presence of hot, coronal gas. Since
models for the WIM with a ~1 kpc scale-height have been widely used to estimate
distances to radio pulsars, our revised parameters suggest that the distances
to many high-latitude pulsars have been substantially underestimated.
