We report the magnetic field dependence of the critical solidification rate for the stability of liquid-solid interfaces. For a certain temperature gradient, the critical solidification rate first increases, then decreases, and subsequently increases with increasing magnetic field. The effect of the magnetic field on the critical solidification rate is more pronounced at low than at high temperature gradients. The numerical simulations show that the magnetic-field dependent changes of convection velocity and contour at the interface agree with the experimental results. The convection velocity first increases, then decreases, and finally increases again with increasing the magnetic field intensity. The variation of the convection contour at the interface first decreases, then increases slightly, and finally increases remarkably with increasing the magnetic field intensity. Thermoelectromagnetic convection (TEMC) plays the role of micro-stirring the melt and is responsible for the increase of interface stability within the initially increasing range of magnetic field intensity. The weak and significant extents of the magneto-hydrodynamic damping (MHD)-dependent solute build-up at the interface front result, respectively, in the gradual decrease and increase of interfacial stability with increasing the magnetic field intensity. The variation of the liquid-side concentration at the liquid-solid interface with the magnetic field supports the proposed mechanism.