A new isomeric 0(+) state was identified as the first excited state in the self-conjugate (N=Z) nucleus 72Kr. By combining for the first time conversion-electron and gamma-ray spectroscopy with the production of metastable states in high-energy fragmentation, the electric-monopole decay of the new isomer to the ground state was established. The new 0(+) state is understood as the band head of the known prolate rotational structure, which strongly supports the interpretation that 72Kr is one of the rare nuclei having an oblate-deformed ground state. This observation gives in fact the first evidence for a shape isomer in a N=Z nucleus.