We study the linear stability of surface-tension-driven unidirectional shear flow in an unbounded electrically conducting liquid layer heated from the side and subjected to a uniform magnetic field in the plane of the layer. The threshold of convective instability with respect to oblique travelling waves is calculated depending on the strength and orientation of the magnetic field. For longitudinal waves the critical Marangoni number and the corresponding wavelength are found to increase directly with the induction of a sufficiently strong magnetic field. In general, a coplanar magnetic field causes stabilization of all disturbances except those aligned with the field, which are not influenced at all. With increase of the magnetic field this effect results in the alignment of the most unstable disturbance along the magnetic flux lines. The maximal stabilization is ensured by the magnetic field being imposed spanwise to the basic flow. The corresponding critical Marangoni number is found to be almost insensitive to the thermal properties of the bottom. The strength of the magnetic field necessary to attain the maximal stabilization for a thermally well-conducting bottom is considerably lower than that for an insulating bottom. The basic return flow is found to be linearly stable with respect to purely hydrodynamic disturbances. This effect determines the stability of the basic state with respect to transverse hydrothermal waves at Prandtl number Pr<Prc=0.018. For such a small Pr no alignment of the critical perturbation with a spanwise magnetic field is possible, and the critical Marangoni number can be increased almost directly with the strength of the magnetic field without limit.