  We determine the Milky Way (MW) mass profile inferred from fitting physically
motivated models to the Gaia DR2 Galactic rotation curve and other data. Using
various hydrodynamical simulations of MW-mass haloes, we show that the presence
of baryons induces a contraction of the dark matter (DM) distribution in the
inner regions, r<20 kpc. We provide an analytic expression that relates the
baryonic distribution to the change in the DM halo profile. For our galaxy, the
contraction increases the enclosed DM halo mass by factors of roughly 1.3, 2
and 4 at radial distances of 20, 8 and 1 kpc, respectively compared to an
uncontracted halo. Ignoring this contraction results in systematic biases in
the inferred halo mass and concentration. We provide a best-fitting contracted
NFW halo model to the MW rotation curve that matches the data very well. The
best-fit has a DM halo mass, $M_{200}^{\rm
DM}=0.97_{-0.19}^{+0.24}\times10^{12} M_\odot$, and concentration before baryon
contraction of $9.4_{-2.6}^{+1.9}$, which lie close to the median halo
mass--concentration relation predicted in $\Lambda$CDM. The inferred total
mass, $M_{200}^{\rm total}=1.08_{-0.14}^{+0.20} \times 10^{12} M_\odot$, is in
good agreement with recent measurements. The model gives a MW stellar mass of
$5.04_{-0.52}^{+0.43}\times10^{10} M_\odot$ and infers that the DM density at
the Solar position is $\rho_{\odot}^{\rm DM}=8.8_{-0.5}^{+0.5}\times10^{-3}
M_\odot \rm{pc}^{-3}\equiv0.33_{-0.02}^{+0.02}~\rm{GeV}~\rm{cm}^{-3}$. The
rotation curve data can also be fitted with an uncontracted NFW halo model, but
with very different DM and stellar parameters. The observations prefer the
physically motivated contracted NFW halo, but the measurement uncertainties are
too large to rule out the uncontracted NFW halo.
