  We study nonlinear bound states, or solitary waves, in the Dirac-Maxwell
system proving the existence of solutions in which the Dirac wave function is
of the form $\phi(x,\omega)e^{-i\omega t}$, $\omega\in(-m,\omega_*)$, with some
$\omega_*>-m$, such that $\phi_\omega\in H^1(\mathbb{R}^3,\mathbb{C}^4)$,
$\Vert\phi_\omega\Vert^2_{L^2}=O(m-|\omega|)$, and
$\Vert\phi_\omega\Vert_{L^\infty}=O(m-|\omega|)$. The method of proof is an
implicit function theorem argument based on an identification of the
nonrelativistic limit as the ground state of the Choquard equation.
