  In this paper we show that conifold transitions between Calabi-Yau 3-folds
can be used for the construction of mirror manifolds and for the computation of
the instanton numbers of rational curves on complete intersection Calabi-Yau
3-folds in Grassmannians. Using a natural degeneration of Grassmannians
$G(k,n)$ to some Gorenstein toric Fano varieties $P(k,n)$ with conifolds
singularities which was recently described by Sturmfels, we suggest an explicit
mirror construction for Calabi-Yau complete intersections $X \subset G(k,n)$ of
arbitrary dimension. Our mirror construction is consistent with the formula for
the Lax operator conjectured by Eguchi, Hori and Xiong for gravitational
quantum cohomology of Grassmannians.
