We extend the principles of cavity quantum electrodynamics to nuclei inside low-loss cavities. The resulting nucleus-cavity system exhibits spontaneous emission and internal conversion rates which differ radically from those of the individual components. The quantum mechanical derivations of these rates are given. The internal conversion process occurs via the exchange of a virtual photon between an excited nucleus and an electron bound to that nucleus. Even though the photon is virtual, partial suppression of the internal conversion process is shown to be theoretically possible. With a controlled modification of the internal conversion rate using a high-Q cavity, one can achieve a number of potentially interesting applications including an excited nuclear state containment unit, a nuclear battery, and a neutron detector.