We have observed microwave spectra of the inert gas diatomic ions He2+, HeAr+, HeKr+ and NeAr+. The spectra arise from transitions between levels which are within a few cm-1 of the lowest dissociation limit. We use mass-selected ion beams which pass first through a waveguide cell, and then through an electric field lens in which selective fragmentation of weakly bound levels occurs. Fragment ions produced in the lens are separated by an electrostatic analyser and detected; spectra are observed by measuring changes in the fragment ion currents. The results are discussed in terms of a theoretical model which relates the high-lying vibration-rotation levels of the diatomic ions to their atomic dissociation limits.