In the rest frame of the Local Group (LG), the total momentum of the Milky Way (MW) and Andromeda (M31) should balance to zero. We use this fact to constrain new solutions for the solar motion with respect to the LG centre-of-mass, the total mass of the LG, and the individual masses of M31 and the MW. Using the set of remote LG galaxies at $>350$ kpc from the MW and M31, we find that the solar motion has amplitude $V_{⊙}=299± 15 {\rm ~km~s^{-1}}$ in a direction pointing toward galactic longitude $l_{⊙}=98.4^{∘}± 3.6^{∘}$ and galactic latitude $b_{⊙}=-5.9^{∘}± 3.0^{∘}$. The velocities of M31 and the MW in this rest frame give a direct measurement of their mass ratio, for which we find $\log_{10} (M_{\rm M31}/M_{\rm MW})=0.36 ± 0.29$. We combine these measurements with the virial theorem to estimate the total mass within the LG as $M_{\rm LG}=(2.5± 0.4)\times 10^{12}~{\rm M}_{⊙}$. Our value for $M_{\rm LG}$ is consistent with the sum of literature values for $M_{\rm MW}$ and $M_{\rm M31}$. This suggests that the mass of the LG is almost entirely located within the two largest galaxies rather than being dispersed on larger scales or in a background medium. The outskirts of the LG are seemingly rather empty. Combining our measurement for $M_{\rm LG}$ and the mass ratio, we estimate the individual masses of the MW and M31 to be $M_{\rm MW}=(0.8± 0.5)\times 10^{12}~{\rm M}_{⊙}$ and $M_{\rm M31}=(1.7± 0.3)\times 10^{12}~{\rm M}_{⊙}$, respectively. Our analysis favours M31 being more massive than the MW by a factor of $∼$2.3, and the uncertainties allow only a small probability (9.8%) that the MW is more massive. This is consistent with other properties such as the maximum rotational velocities, total stellar content, and numbers of globular clusters and dwarf satellites, which all suggest that $M_{\rm M31}/M_{\rm MW}>1$. ; Comment: 16 pages, 11 figures, 3 tables. Accepted for publication in MNRAS