We report the magnetic field dependence of the low temperature specific heat of single crystals of the first Pr-based heavy fermion superconductor PrOs4Sb12. 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 PrOs4Sb12 as a very strong contender of superconductive pairing that is neither electron-phonon nor magnetically mediated.PACS numbers: 74.70. Tx, 7127,+a, 75.30.Mb Superconductivity mediated by a pairing potential other than a conventional electron-phonon interaction has been the subject of a very large number of theoretical and experimental investigations over the decades. In recent years, finally, intermetallic compounds have been discovered which, together with the high-T c cuprates, represent prime candidates for magnetically mediated pairing. Surprisingly, however, magnetically mediated pairing thus far has been considered the only serious alternative to electron-phonon mediated pairing, while excitonic and polaronic mechanisms have also been proposed.We have recently reported the discovery of the first Pr-based heavy fermion superconductor PrOs 4 Sb 12 [1], for which the nonmagnetic ground state appeared best described as a crystalline electric field (CEF) doublet. This in turn suggested that the heavy electron liquid is of quadrupolar origin and that consequently quadrupolar fluctuations mediate the superconductive pairing. PrOs 4 Sb 12 hence is a candidate for being the first material in which neither electron-phonon nor magnetic interactions mediate the pairing. However, a magnetic origin could not be completely ruled out [1] and recent low temperature specific heat measurements are reported to be consistent with a singlet CEF ground state hence questioning the hypothesis of quadrupolar pairing [2].In order to settle the question of the ground state and thus the possibility of the first example of quadrupolar mediated superconductive pairing demands definitive establishement of (i) the CEF level scheme, (ii) the unconventional nature of the superconductivity, and (iii) coupling of the CEF excitations to the conduction electrons. Here we report specific heat measurements of high quality single crystals of PrOs 4 Sb 12 at low T and high magnetic field. We show that the zero field data of the single crystals and the magnetic phase diagram as established from the specific heat, resistivity and magnetisation, as well as the observation of a novel high field ground state, provide compelling evidence of a nonmagnetic doublet CEF ground state intimately linked to the conduction electrons. This unambiguously establishes quadrupolar fluctuations as the most likely pairing mechanism. Moreover, we observe two superconducting transitions giving evidence of two distinct superconducting phases.Previous experiments on pressed pellets of tiny single crystals of PrOs 4 Sb 12 [1] revealed superconductivity ...