The vacuum fluctuations of the photon and pair fields modify the interaction of an electron with an electromagnetic field. The effects on the energy levels are conveniently described in terms of the mass operator and the vacuum polarization potential. A gauge-covariant expansion of the mass operator for the motion of an electron in a weak external electromagnetic field is derived; the expression contains terms quadratic in the field but includes only the lowest order electrodynamic correction. The modification in the Fermi formula is then computed by specializing the external field to consist of the Coulomb and magnetic dipole fields of the nucleus and by taking the matrix element of the operators in an $S$-state of a hydrogen-like atom. All changes can be described as a correction $\ensuremath{\Delta}g=\ensuremath{-}2Z{\ensuremath{\alpha}}^{2}(\frac{5}{2}\ensuremath{-}\mathrm{ln}2)$ in the gyromagnetic ratio of the electron. The value of the fine structure constant deduced from measurements of the hyperfine structure becomes ${\ensuremath{\alpha}}^{\ensuremath{-}1}=137.0364$.