  We investigate the L\'evy glass, a mean-field spin glass model with power-law
distributed couplings characterized by a divergent second moment. By combining
extensively many small couplings with a spare random backbone of strong bonds
the model is intermediate between the Sherrington-Kirkpatrick and the
Viana-Bray model. A truncated version where couplings smaller than some
threshold $\eps$ are neglected can be studied within the cavity method
developed for spin glasses on locally tree-like random graphs. By performing
the limit $\eps\to 0$ in a well-defined way we calculate the thermodynamic
functions within replica symmetry and determine the de Almeida-Thouless line in
the presence of an external magnetic field. Contrary to previous findings we
show that there is no replica-symmetric spin glass phase. Moreover we determine
the leading corrections to the ground-state energy within one-step replica
symmetry breaking. The effects due to the breaking of replica symmetry appear
to be small in accordance with the intuitive picture that a few strong bonds
per spin reduce the degree of frustration in the system.
