Constant-density electrically conducting fluid is confined to a rapidly rotating spherical shell and is permeated by an axisymmetric potential magnetic field with dipole parity; the regions outside the shell are rigid insulators. Slow steady axisymmetric motion is driven by rotating the inner sphere at a slightly slower rate. Linear solutions of the governing magnetohydrodynamic equations are derived in the small Ekman number E-limit for values of the Elsasser number Λ less than order unity. Attention is restricted to the mainstream outside the Ekman–Hartmann layers adjacent to the inner and outer boundaries.