The silicon clathrates-materials composed of metal-doped Si-20 dodecahedra-were identified as the first superconductors based on pure silicon networks(1,2). The mechanism of superconductivity in these materials can be obtained by studying their phonon modes, as modified by isotope substitution, and specific-heat measurements. Here, we present experimental studies that provide strong evidence that superconductivity in Ba8Si46 is explained in the framework of phonon-mediated Bardeen-Cooper-Schriefer theory. Analyses using the McMillan approximation(3,4) of the Eliashberg equation indicate that the superconducting mechanism is in the medium coupling regime, but at the high-end limit. The large density of states at the Fermi level, which arises from hybridization of the Si-20 cluster and Ba orbitals, is responsible for the unexpectedly high superconducting temperature. The temperature evolution of the specific heat unambiguously shows that this is an s-wave symmetry superconductor.