  We present an experimental concept and setup for laser-microwave
double-resonance spectroscopy of highly charged ions in a Penning trap. Such
spectroscopy allows a highly precise measurement of the Zeeman splittings of
fine- and hyperfine-structure levels due the magnetic field of the trap. We
have performed detailed calculations of the Zeeman effect in the framework of
quantum electrodynamics of bound states as present in such highly charged ions.
We find that apart from the linear Zeeman effect, second- and third-order
Zeeman effects also contribute to the splittings on a level of 10^-4 and 10^-8,
respectively, and hence are accessible to a determination within the achievable
spectroscopic resolution of the ARTEMIS experiment currently in preparation.
