A study of the effect of an externally imposed magnetic field on the axisymmetry-breaking instability of an axisymmetric convective flow, associated with crystal growth from bulk of melt, is presented. Convection in a vertical cylinder with a parabolic temperature profile on the sidewall is considered as a representative model. A parametric study of the dependence of the critical Grashof number Gr(cr) on the Hartmann number Ha for fixed values of the Prandtl number (Pr = 0.015) and the aspect ratio of the cylinder (A = height/radius = 1,2 and 3) is carried out. The stability diagram Gr(cr)(Ha) corresponding to the axisymmetric-three-dimensional transition for increasing values of the axial magnetic field is obtained. The calculations are done using the spectral Galerkin method allowing an effective and accurate three-dimensional stability analysis, it is shown that at relatively small values of Ha the axisymmetric flow tends to be oscillatory unstable. After the magnitude of the magnetic field (Ha) exceeds a certain value the instability switches to a steady bifurcation caused by the Rayleigh-Benard mechanism.