This paper treats the steady axisymmetric flow and mass transport in a cylindrical liquid bridge between the melting end of a feed rod and the solidifying end of an alloyed semiconductor crystal, There is a strong, uniform, steady, axial magnetic field. The surface tension depends on the temperature and the concentration of the species, while variations of the concentration occur because one species is rejected into the liquid during solidification. The thermocapillary and solutocapillary convections tend to cancel over part of the liquid bridge, For certain parameter ranges, there are two different stable solutions: one where the concentration gradient along the free surface leads to dominance by the solutocapillary convection and one where the mass transport due to the thermocapillary convection makes the concentration gradient along the free surface small, so that the thermocapillary convection is dominant.