We have observed a microwave spectrum of the HeKr+ ion in which all of the observed levels lie within a few cm(-1) of either the first or second dissociation limit. We use an ion beam technique in which HeKr+ ions, formed by electron impact, are mass analyzed. Passage of the ion beam through an electric field lens results in selective fragmentation of energy levels lying close to dissociation. Kr+ ions formed in the lens are separated from all other ions by means of an electrostatic analyzer, and are detected with an electron multiplier. Microwave radiation induces transitions which result in population transfer and produce detected changes in the electric field-induced Kr+ fragment ion current. Additional transitions have been defected by a microwave-microwave double resonance method, and we have also made extensive use of the Zeeman effects produced by small applied coaxial magnetic fields to identify the J quantum numbers of the levels involved. Coupled channel calculations of the bound states of the He ... Kr+ ion are carried out, fully including all the couplings between different electronic sates correlating with He+Kr+ (P-2(3/2) and P-2(1/2)). The calculations allow the spectra to be assigned to pure rotational transitions involving levels in the X, A(1), and A(2) states that lie within 2.5 cm(-1) of the dissociation limits. Because of a systematic near degeneracy between vibrational levels in the X and A(1) stares, the long-range He ... Kr+ ion provides a very good example of Hund’s case (e) in the form introduced by Mulliken, in which then are no projection quantum numbers onto the interatomic axis. Mulliken’s case (e) is rather different from the Rydberg case (e) described by Lefebvre-Brion, and this is the first time that Mulliken’s case (e) has been observed. The spectra allow the interaction potential for He ... Kr+ to be determined accurately, for the first time, by least-squares fitting of potential parameters to the experimental Tine frequencies and g factors. The resulting interaction potential (designated MAL1) is compared with that previously determined for He ... Ar+ : the He ... Kr+ potential is significantly shallower, because the long-range ion-induced dipole C-4 coefficient is the same for the two systems but the larger Kr+ ion prevents the He atom approaching as close.