  We present an {\it ab initio} study of the recently discovered
superconductivity of boron doped diamond within the framework of a
phonon-mediated pairing mechanism. The role of the dopant, in substitutional
position, is unconventional in that half of the coupling parameter $\lambda$
originates in strongly localized defect-related vibrational modes, yielding a
very peaked Eliashberg $\alpha^2F(\omega)$ function. The electron-phonon
coupling potential is found to be extremely large and T$_C$ is limited by the
low value of the density of states at the Fermi level.
