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We estimate the mass of the inner ($<20$ kpc) Milky Way and the axis ratio ofits dark matter halo using globular clusters as tracers. At the same time, weconstrain the phase-space distribution of the globular cluster system. We usethe Gaia DR2 catalogue of 75 globular clusters' proper motions and recentmeasurements of the proper motions of another 20 distant clusters obtained withthe Hubble Space Telescope. We describe the globular cluster system with a2-component distribution function (DF), with a flat, rotating disc and arounder, more extended halo. While fixing the Milky Way's disc and bulge, welet the mass and shape of the dark matter halo and we fit these two parameters,together with other six describing the DF, with a Bayesian method. We find themass of the Galaxy within 20 kpc to be $M(<20{\,\rm kpc})=1.91^{+0.18}_{-0.17}\times 10^{11} M_\odot$, of which $M_{\rm DM}(<20{\,\rmkpc})=1.37^{+0.18}_{-0.17}\times 10^{11}M_\odot$ is in dark matter, and thedensity axis ratio of the dark matter halo to be $q=1.30 \pm 0.25$. Thisimplies a virial mass $M_{\rm via} = 1.3 \pm 0.3 \times 10^{12} M_\odot$. Ouranalysis rules out oblate ($q<0.8$) and strongly prolate halos ($q>1.9$) with99\% probability. Our preferred model reproduces well the observed phase-spacedistribution of globular clusters and has a disc component that closelyresembles that of the Galactic thick disc. The halo component follows apower-law density profile $\rho \propto r^{-3.3}$, has a mean rotationalvelocity of $V_{\rm rot}\simeq -14\,\rm km\,s^{-1}$ at 20 kpc, and has a mildlyradially biased velocity distribution ($\beta\simeq 0.2 \pm 0.07$, fairlyconstant outside 15 kpc). We also find that our distinction between disc andhalo clusters resembles, although not fully, the observed distinction inmetal-rich ([Fe/H]$>-0.8$) and metal-poor ([Fe/H]$\leq-0.8$) clusterpopulations.

Mass and shape of the Milky Way's dark matter halo with globular clusters from Gaia and Hubble