A precision measurement of the $Bc+$ meson mass is performed using proton-proton collision data collected with the LHCb experiment at centre-of-mass energies of $7,8$ and $13$ TeV, corresponding to a total integrated luminosity of $9.0fb-1$. The $Bc+$ mesons are reconstructed via the decays $Bc+\to J/\psi \pi +$, $Bc+\to J/\psi \pi +\pi -\pi +$, $Bc+\to J/\psi pp¯\pi +$, $Bc+\to J/\psi Ds+$, $Bc+\to J/\psi D0K+$ and $Bc+\to Bs0\pi +$. Combining the results of the individual decay channels, the $Bc+$ mass is measured to be $6274.47\pm 0.27(stat)\pm 0.17(syst)\mathrm{Me}V/c2$. This is the most precise measurement of the $Bc+$ mass to date. The difference between the $Bc+$ and $Bs0$ meson masses is measured to be $907.75\pm 0.37(stat)\pm 0.27(syst)\mathrm{Me}V/c2$.