We report the magnetic field dependence of the specific-heat C of single crystals of the first Pr-based heavy-fermion superconductor Pr(Os4Sb12. The variation of C at low temperature and the magnetic phase diagram inferred from C, the resistivity and magnetization provide compelling evidence of a doublet ground state. Two distinct superconducting anomalies in C indicate an unconventional superconducting state, where the splitting may arise from a weak lifting of the ground state degeneracy. In combination this identifies Pr(Os4Sb12 as a strong contender for quadrupolar pairing, i.e., superconductivity that is neither electron-phonon nor magnetically mediated.