This paper treats the motion and heat transfer in an electrically conducting fluid that is bounded by two parallel, isothermal walls, and by a plane free surface. Gradients in the surface tension of the free surface drive a strong recirculating thermocapillary convection. In the presence of a magnetic field which is parallel to the free surface, the thermocapillary convection is confined to a thin boundary layer adjacent to the free surface. Only when the Peclet number is at least O(Ha(1/4)), where Ha is the large Hartmann number, is the heat flux entering the free surface redistributed by thermal convection inside the boundary layer before entering the quiescent core. This problem is a model of the floating-zone growth of semiconductor crystals with a uniform, axial magnetic field.