We discuss the application of Gutzwiller's semiclassical theory to a circular billiard with a singular magnetic flux line added at its center. The Aharonov-Bohm effect manifests itself through the cancellation of periodic orbits for particular field strengths. Diffraction phenomena affect the gross-shell structure of the level density and require corrections of higher order in \hbar. The full quantization of the level spectrum, however, is much less affected.