We construct a model for the Milky Way Galaxy composed of a stellar disc and bulge embedded in a dark-matter halo. All components are modelled as $N$-body systems with up to 8 billion equal-mass particles and integrated up to an age of 10\,Gyr. We find that net angular-momentum of the dark-matter halo with a spin parameter of $λ=0.06$ is required to form a relatively short bar ($\sim 4$\,kpc) with a high pattern speed (40--50\,km\,s$^{-1}$). By comparing our model with observations of the Milky Way Galaxy, we conclude that a disc mass of $\sim 3.7\times10^{10}M_{\odot}$ and an initial bulge scale length and velocity of $\sim 1$\,kpc and $\sim 300$\,km\,s$^{-1}$, respectively, fit best to the observations. The disc-to-total mass fraction ($f_{\rm d}$) appears to be an important parameter for the evolution of the Galaxy and models with $f_{\rm d}\sim 0.45$ are most similar to the Milky Way Galaxy. In addition, we compare the velocity distribution in the solar neighbourhood in our simulations with observations in the Milky Way Galaxy. In our simulations the observed gap in the velocity distribution, which is expected to be caused by the outer Lindblad resonance (the so-called Hercules stream), appears to be a time-dependent structure. The velocity distribution changes on a time scale of 20--30\,Myr and therefore it is difficult to estimate the pattern speed of the bar from the shape of the local velocity distribution alone.

Accepted by MNRAS, 32 pages, 31 figures