We study a fully connected spin-glass model in which the coupling strengths are drawn from a Lévy distribution. In this model each spin can be seen as interacting through a finite number of strong bonds and an infinite number of weak bonds, due to the large tail behavior of the coupling distribution. We solve the model through the replica and the cavity method. The hybrid behavior of Lévy spin glasses becomes transparent in our solution: the local field contains a part propagating along a backbone of strong bonds and a Gaussian noise term due to weak bonds. Our method allows us to determine the complete replica symmetric phase diagram, the replica symmetry breaking line and the entropy. The results are compared with ones from simulations and previous calculations using a Gaussian ansatz for the distribution of fields.