  We study the behavior of a single laser-driven trapped ion inside a
microscopic optical Fabry-Perot cavity. In particular, we demonstrate a fiber
Fabry-Perot cavity operating on the principal $S_{1/2}\to P_{1/2}$ electric
dipole transition of an Yb$^+$ ion at $369\,$nm with an atom-ion coupling
strength of $g=2\pi\times 67(1)\,$MHz. We employ the cavity to study the
generation of single photons and observe cavity-induced back-action in the
Purcell-enhanced emission of photons. Tuning of the amplitude and phase of the
back-action allows us to enhance or suppress the total rate of photoemission
from the ion-cavity system.
