  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.
