  The effect of a finite nuclear mass on the magnetic moment of the electron
bound in the ground state of a hydrogen-like ion is analyzed. Using the exact
in Z\alpha expression for the recoil shift of the energy, I calculate the order
(Z\alpha)^4 m/M correction to the electron gyromagnetic ratio. The theoretical
uncertainty in the prediction for the experimentally interesting carbon ion is
decreased to the level of 1.2x10^-9. This provides a factor of two improvement
in the precision of the atomic mass of the electron.
