  Context. Gas within a galaxy is forced to establish pressure balance against
gravitational forces. The shape of an unperturbed gaseous disk can be used to
constrain dark matter models. Aims. We derive the 3-D HI volume density
distribution for the Milky Way out to a galactocentric radius of 40 kpc and a
height of 20 kpc to constrain the Galactic mass distribution. Methods. We used
the Leiden/Argentine/Bonn all sky 21-cm line survey. The transformation from
brightness temperatures to densities depends on the rotation curve. We explored
several models, reflecting different dark matter distributions. Each of these
models was set up to solve the combined Poisson-Boltzmann equation in a
self-consistent way and optimized to reproduce the observed flaring. Results.
Besides a massive extended halo of M ~ 1.8 10^{12} Msun, we find a
self-gravitating dark matter disk with M=2 to 3 10^{11} Msun, including a dark
matter ring at 13 < R < 18.5 kpc with M = 2.2 to 2.8 10^{10} Msun. The
existence of the ring was previously postulated from EGRET data and coincides
with a giant stellar structure that surrounds the Galaxy. The resulting Milky
Way rotation curve is flat up to R~27 kpc and slowly decreases outwards. The
\hi gas layer is strongly flaring. The HWHM scale height is 60 pc at R = 4 kpc
and increases to ~2700$ pc at R=40 kpc. Spiral arms cause a noticeable imprint
on the gravitational field, at least out to R = 30 kpc. Conclusions. Our mass
model supports previous proposals that the giant stellar ring structure is due
to a merging dwarf galaxy. The fact that the majority of the dark matter in the
Milky Way for $R \la 40$ kpc can be successfully modeled by a self-gravitating
isothermal disk raises the question of whether this massive disk may have been
caused by similar merger events in the past.
