  Let $X$ be the product of two projective spaces and consider the general CICY
threefold $Y$ in $X$ with configuration matrix $A$. We prove the finiteness
part of the analogue of the Clemens' conjecture for such a CICY in low
bidegrees. More precisely, we prove that the number of smooth rational curves
on $Y$ with low bidegree and with nondegenerate birational projection is at
most finite (even in cases in which positive dimensional families of degenerate
rational curves are known).
