  In this paper, we present an explicit structure for the Baer invariant of a
free nilpotent group (the $n$-th nilpotent product of the infinite cyclic
group, $\textbf{Z}\st{n}* \textbf{Z}\st{n}*... \st{n}*\textbf{Z}$) with respect
to the variety of polynilpotent groups of class row $(c,1)$, ${\cal N}_{c,1}$,
for all $c > 2n-2$. In particular, an explicit structure of the Baer invariant
of a free abelian group with respect to the variety of metabelian groups will
be presented.
