  Fast radio bursts offer the opportunity to place new constraints on the mass
and density profile of hot and ionized gas in galactic haloes. We test here the
X-ray emission and dispersion measure predicted by different gas profiles for
the halo of the Milky Way. We examine a range of models, including entropy
stability conditions and external pressure continuity. We find that
incorporating constraints from X-ray observations leads to favouring dispersion
measures on the lower end of the range given by these models. We show that the
dispersion measure of the Milky Way halo could be less than 10 cm$^{-3}$ pc in
the most extreme model we consider, which is based on constraints from O VII
absorption lines. However, the models allowed by the soft X-ray constraints
span more than an order of magnitude in dispersion measures. Additional
information on the distribution of gas in the Milky Way halo could be obtained
from the signature of a dipole in the dispersion measure of fast radio bursts
across the sky, but this will be a small effect for most cases.
