  The excited states of a charged particle interacting with the quantized
electromagnetic field and an external potential all decay, but such a particle
should have a true ground state--one that minimizes the energy and satisfies
the Schr\"odinger equation. We prove quite generally that this state exists for
{\it all values} of the fine-structure constant and ultraviolet cutoff. We also
show the same thing for a many-particle system under physically natural
conditions.
