The theory of octupolar-excitation ion-cyclotron-resonance mass spectrometry is presented which predicts an increase of up to several orders of magnitude in resolving power under certain conditions. The new method has been applied for a direct Penning-trap mass-ratio determination of the Er164-Dy164 mass doublet. Er164 is a candidate for the search for neutrinoless double-electron capture. However, the measured Q value of 25.07(12) keV results in a half-life of 1030 years for a 1 eV Majorana-neutrino mass.