  We study the behavior of the spectrum of the Dirac operator together with a
symmetric $W^{1, \infty}$-potential on spin manifolds under a collapse of
codimension one with bounded sectional curvature and diameter. If there is an
induced spin structure on the limit space $N$ then there are convergent
eigenvalues which converge to the spectrum of a first order differential
operator $D$ on $N$ together with a symmetric $W^{1,\infty}$-potential. If $N$
is orientable and the dimension of the limit space is even then $D$ is the
Dirac operator $D^N$ on $N$ and if the dimension of the limit space is odd,
then $D = D^N \oplus -D^N$.
