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 (164)Er-(164)Dy mass doublet. (164)Er is a candidate for the search for neutrinoless double-electron capture. However, the measured Q(ϵϵ) value of 25.07(12) keV results in a half-life of 10(30) years for a 1 eV Majorana-neutrino mass.