The problem of an electron-hole system interacting through a contact potential and moving in a one-dimensional quantum ring threaded by an Aharonov-Bohm flux is considered, with respect to the system's energetics as well as its optical properties. An exact analytical expression for the energy spectrum is derived using a straightforward method based on boundary conditions for wavefunctions and their derivatives along the ring. The optical properties of this exciton system, namely linear dynamic polarizability and the absorption spectrum, are investigated and certain unusual features are demonstrated. It is shown, for example, that for special values of the magnetic flux there are energies in the spectrum that correspond to the dark excitonic states.
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