  Using the Pauli-Fierz model of non-relativistic quantum electrodynamics, we
calculate the binding energy of an electron in the field of a nucleus of charge
$Z$ and in presence of the quantized radiation field. We consider the case of
small coupling constant $\alpha$, but fixed $Z\alpha$ and ultraviolet cut-off
$\Lambda$. We prove that after renormalizing the mass the binding energy has,
to leading order in $\alpha$, a finite limit as $\Lambda$ goes to infinity;
i.e., the cut-off can be removed. The expression for the ground state energy
shift thus obtained agrees with Bethe's formula for small values of $Z\alpha$,
but shows a different behavior for bigger values.
