  We prove the rationality of the K\"ahler cone and the positivity of $c_2(X)$,
if $X$ is a Calabi-Yau-threefold with $\rho(X)=2$ and has an embedding into a
${\bb P}^n$-bundle over ${\bb P}^m$ in the cases $(n,m)=(1,3),(3,1)$. The case
$(n,m)=(2,2)$ has been done in the first part of this paper. Moreover, if
$(n,m)=(3,1)$, we describe the 'other' contraction different from the
projection.
