  We investigate a recent proposal to construct chiral gauge theories on the
lattice using domain wall fermions. We restrict ourselves to the finite volume
case, in which two domain walls are present, with modes of opposite chirality
on each of them. We couple the chiral fermions on only one of the domain walls
to a gauge field. In order to preserve gauge invariance, we have to add a
scalar field, which gives rise to additional light mirror fermion and scalar
modes. We argue that in an anomaly free model these extra modes would decouple
if our model possesses a so-called strong coupling symmetric phase. However,
our numerical results indicate that such a phase most probably does not exist.
---- Note: 9 Postscript figures are appended as uuencoded compressed tar file.
