  The energy-level structure of a single atom strongly coupled to the mode of a
high-finesse optical cavity is investigated. The atom is stored in an
intracavity dipole trap and cavity cooling is used to compensate for inevitable
heating. Two well-resolved normal modes are observed both in the cavity
transmission and the trap lifetime. The experiment is in good agreement with a
Monte Carlo simulation, demonstrating our ability to localize the atom to
within $\lambda/10$ at a cavity antinode.
