Abstract: We have carried out a high-resolution spectroscopic study of the absorption of submillimeter radiation by free excitons in germanium compressed along the [ 1 11 ] axis in a magnetic field parallel to the compression axis. In particular, we studied the splitting of the 1s- 2p transition in fields up to 6 kOe at T = 1.6 K, and observed a complex pattern in the Zeeman splitting which we believe is related to the effect of thermal motion of the excitons in a magnetic field on their internal structure (the magneto-Stark effect). The calculated submillimeter spectrum of excitons agrees with the experimental data. We predict that in a magnetic field the energy of the 2p, term is a minimum at a finite value of the exciton momentum perpendicular to the field-that is, the energy minimum forms a ring in momentum space. It follows that the density of states for this term must be a nonmonotonic function of the energy. A theory is developed of analogous phenomena in positronium.