  We have carried out a numerical simulation of a domain-wall model in
$(2+1)$-dimensions, in the presence of a dynamical gauge field only in an extra
dimension, corresponding to the weak coupling limit of a ( 2-dimensional )
physical gauge coupling. Using a quenched approximation we have investigated
this model at $\beta_{s} ( = 1 / g^{2}_{s} ) =$ 0.5 ( ``symmetric'' phase),
1.0, and 5.0 (``broken'' phase), where $g_s$ is the gauge coupling constant of
the extra dimension. We have found that there exists a critical value of a
domain-wall mass $m_{0}^{c}$ which separates a region with a fermionic zero
mode on the domain-wall from the one without it, in both symmetric and broken
phases. This result suggests that the domain-wall method may work for the
construction of lattice chiral gauge theories.
