We present the design and implementation of a trapped ion-cavity QED system. A single ytterbium ion is confined by a micron-scale ion trap inside a 2-mm optical cavity. The ion is coherently pumped by near-resonant laser light while the cavity output is monitored as a function of pump intensity and cavity detuning. We observe a Purcell enhancement of scattered light into the solid angle subtended by the optical cavity, as well as a three-peak structure arising from strongly driving the atom. This system can be integrated into existing atom-photon quantum network protocols and is a pathway toward an efficient atom-photon quantum interface.