  In the Nelson model particles interact through a scalar massless field. For
hydrogen-like atoms there is a nucleus of infinite mass and charge $Ze$, $Z >
0$, fixed at the origin and an electron of mass $m$ and charge $e$. This system
forms a bound state with binding energy $E_{\rm bin} = me^4Z^2/2$ to leading
order in $e$. We investigate the radiative corrections to the binding energy
and prove upper and lower bounds which imply that $ E_{\rm bin} = me^4 Z^2/2 +
c_0 e^6 + \Ow(e^7 \ln e)$ with explicit coefficient $c_0$ and independent of
the ultraviolet cutoff. $c_0$ can be computed by perturbation theory, which
however is only formal since for the Nelson Hamiltonian the smallest eigenvalue
sits exactly at the bottom of the continuous spectrum.
