  The energy level of the exciton around the $\Gamma$ point in CaF$_2$ at about
11.1 eV was investigated by reflection measurements on oriented surfaces using
dispersed synchrotron radiation. A small but clear energy shift of the exciton
resonance structure was observed which depends on the crystal orientation with
respect to the incoming beam. By interpreting the results in terms of the
complex dielectric function it becomes possible to determin the respective
exciton levels and lifetimes. For crystal samples oriented in the (111)
direction, the exciton has an excitation energy 0.02 eV above the energy of
samples oriented in the (100) direction. The results obtained suggest that the
exciton-energy shift is related to the spatial anisotropy recently measured in
CaF$_2$ at 157 nm.
