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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.