  We report the magnetic field dependence of the low temperature specific heat
of single crystals of the first Pr-based heavy fermion superconductor
PrOs$_4$Sb$_{12}$. The low temperature specific heat and the magnetic phase
diagram inferred from specific heat, resistivity and magnetisation provide
compelling evidence of a doublet ground state and hence superconductivity
mediated by quadrupolar fluctuations. This establishes PrOs$_4$Sb$_{12}$ as a
very strong contender of superconductive pairing that is neither
electron-phonon nor magnetically mediated.
