Measurements made with a dilute, non-oriented, gas-phase sample of a selected fenchone enantiomer using circularly polarized synchrotron radiation demonstrate huge chiral asymmetries, approaching 20%, in the anaular distribution of photoelectrons ejected from carbonyl C 1s core orbitals. This asymmetry in the forward-backward scattering of electrons along the direction of the incident soft X-ray radiation reverses when either the enantiomer or the left-right handedness of the light polarization is exchanged. Calculations are provided that model and explain the resulting photoelectron circular dichroism with quantitative accuracy up to -7 eV above threshold. A discrepancy at higher energies is discussed in the light of a comparison with the closely related terpene, camphor. The photoelectron dichroism spectrum can be used to identify the absolute chiral configuration, and it is more effective at distinguishing the similar camphor and fenchone molecules than the corresponding core photoelectron spectrum.