  We consider a hydrogen-like atom in a quantized electromagnetic field which
is modeled by means of the semi-relativistic Pauli-Fierz operator and prove
that the infimum of the spectrum of the latter operator is an eigenvalue. In
particular, we verify that the bottom of its spectrum is strictly less than its
ionization threshold. These results hold true for arbitrary values of the
fine-structure constant and the ultra-violet cut-off as long as the Coulomb
coupling constant (i.e. the product of the fine-structure constant and the
nuclear charge) is less than 2/\pi.
