  The distribution of shapes of galaxies' dark halos provides a basic test for
models of galaxy formation. To-date, few dark halo shapes have been measured,
and the results of different methods appear contradictory. Here, we add to the
sample of measured shapes by calculating the flattening of the Milky Way's dark
halo based on the manner in which the gas layer in the Galaxy flares with
radius. We also test the validity of this technique -- which has already been
applied to several other galaxies -- by comparing the inferred halo flattening
to that obtained from a stellar-kinematic analysis, which can only be applied
to the Milky Way. Both methods return consistent values for the shape of the
Milky Way's halo, with a shortest-to-longest axis ratio for the dark matter
distribution of q = 0.75 +/- 0.25. However, this consistency is only achieved
if we adopt a value of R_0 = 7 +/- 1 kpc for the Sun's distance to the Galactic
center. Although this value is smaller than the IAU-sanctioned R_0 = 8.5 kpc,
it is quite consistent with current observations. Whatever value of R_0 is
adopted, neither method returns halo parameters consistent with a disk-like
mass distribution, for which q ~ 0.2. This finding rules out cold molecular gas
and decaying massive neutrinos as dark matter candidates.
