We consider the signatures of a domain wall produced in the spontaneous symmetry breaking involving a dilaton-like scalar field coupled to electromagnetism. Domains on either side of the wall exhibit slight differences in their respective values of the fine-structure constant, alpha. If such a wall is present within our Hubble volume, absorption spectra at large redshifts may or may not provide a variation in alpha relative to the terrestrial value, depending on our relative position with respect to the wall. This wall could resolve the ``contradiction'' between claims of a variation of alpha based on Keck/Hires data and of the constancy of alpha based on VLT data. We derive the properties of the wall and the parameters of the underlying microscopic model required to reproduce the possible spatial variation of alpha. We discuss the constraints on the existence of the low-energy domain wall and describe its observational implications concerning the variation of the fundamental constants.