  We consider N=2 supersymmetric quantum electrodynamics (SQED) with 2 flavors,
the Fayet--Iliopoulos parameter, and a mass term $\beta$ which breaks the
extended supersymmetry down to N=1. The bulk theory has two vacua; at $\beta=0$
the BPS-saturated domain wall interpolating between them has a moduli space
parameterized by a U(1) phase $\sigma$ which can be promoted to a scalar field
in the effective low-energy theory on the wall world-volume. At small
nonvanishing $\beta$ this field gets a sine-Gordon potential. As a result, only
two discrete degenerate BPS domain walls survive. We find an explicit solitonic
solution for domain lines -- string-like objects living on the surface of the
domain wall which separate wall I from wall II. The domain line is seen as a
BPS kink in the world-volume effective theory. We expect that the wall with the
domain line on it saturates both the $\{1,0\}$ and the $\{{1/2},{1/2}\}$b
central charges of the bulk theory. The domain line carries the magnetic flux
which is exactly 1/2 of the flux carried by the flux tube living in the bulk on
each side of the wall. Thus, the domain lines on the wall confine charges
living on the wall, resembling Polyakov's three-dimensional confinement.
