  We show that the Hamiltonian describing N nonrelativistic electrons with
spin, interacting with the quantized radiation field and several fixed nuclei
with total charge Z has a ground state when N <Z+1. The result holds for any
value of the fine structure constant alpha and for any value of the ultraviolet
cutoff Lambda on the radiation field. There is no infrared cutoff. The basic
mathematical ingredient in our proof is a novel way of localizing the
electromagnetic field in such a way that the errors in the energy are of
smaller order than 1/L, where L is the localization radius.
