This paper deals with the linear stability analysis of the convective flow of an electrically conducting fluid in a spherical gap in the presence of an axial magnetic field which is parallel to the vector of gravitational acceleration. The inner shell is warmer then the outer one (T-1 > T-2). The numerical investigations are performed for the radii ratios eta = R-1/R-2 = 0.4-0.8. The corresponding stability diagrams, i.e. the critical values of the Grashof number Gr(c) and the wave number m(c), are presented in dependence on the Hartmann number Ha. We show that the critical Grashof number decreases with increasing eta in the case of absence of the magnetic field. A steady axial magnetic field stabilizes the flow, i.e. Gr(c) increases with the Hartmann number Ha for each eta investigated. The instability sets in either as a Hopf bifurcation or a steady pitchfork bifurcation in dependence on eta and Ha. The stability analysis is accompanied by simulation of the three-dimensional states. We found that according to the bifurcation type there are two classes of 3D states: steady and oscillatory. Consequences of the symmetry with respect to +/-phi direction are discussed. (C) 2012 Elsevier Ltd. All rights reserved.