  The transmission spectrum for one atom strongly coupled to the field of a
high-finesse optical resonator is observed to exhibit a clearly resolved
vacuum-Rabi splitting characteristic of the normal modes in the eigenvalue
spectrum of the atom-cavity system. A new Raman scheme for cooling atomic
motion along the cavity axis enables a complete spectrum to be recorded for an
individual atom trapped within the cavity mode, in contrast to all previous
measurements in cavity QED that have required averaging over many atoms.
