  We study the interaction between stable monopoles and domain walls in a
$SO(3) \times Z(2)$ scalar field theory. Numerical simulations reveal that
there is an attractive force between the monopole and the wall, but that after
the monopole and the wall collide, the monopole does not unwind. We present an
analytic explanation for the origin of the attractive force, and conclude that
this is a generic feature of monopole-wall interactions which does not depend
on the detailed structure of the model. The existence of the attractive force
supports the hypothesis of Dvali {\em et al.} (hep-ph/9710301), who proposed
that monopoles can be ``swept up'' by domain walls, thereby alleviating or
solving the monopole problem associated with phase transitions occurring after
or in the absence of inflation.
