We present a method of laser frequency stabilization based on the linear dichroism signal in a transverse magnetic field. This method is similar to the DAVLL (Dichroic Atomic Vapor Laser Lock) method. It differs from DAVLL and from its existing modifications primarily by the fact that it uses signals of linearly polarized light caused by alignment, rather than circular refraction caused by orientation, and therefore allows us to obtain error signals at the magnetic field modulation frequency (or its second harmonic) by extremely simple means. The method allows the laser frequency to be stabilized in the vicinity of the low-frequency transition in the D1 line of Cs; it does not require strong magnetic fields or careful shielding of cells containing cesium atoms. Although the absorption line in a gas-filled cell is typically gigahertz wide, the achievable resolution, limited by the signal-to-noise ratio of photon shot noise, can reach units or tens of kilohertz in a one hertz bandwidth.