  We show that the moduli space of all Calabi-Yau manifolds that can be
realized as hypersurfaces described by a transverse polynomial in a four
dimensional weighted projective space, is connected. This is achieved by
exploiting techniques of toric geometry and the construction of Batyrev that
relate Calabi-Yau manifolds to reflexive polyhedra. Taken together with the
previously known fact that the moduli space of all CICY's is connected, and is
moreover connected to the moduli space of the present class of Calabi-Yau
manifolds (since the quintic threefold P_4[5] is both CICY and a hypersurface
in a weighted P_4, this strongly suggests that the moduli space of all simply
connected Calabi-Yau manifolds is connected. It is of interest that singular
Calabi-Yau manifolds corresponding to the points in which the moduli spaces
meet are often, for the present class, more singular than the conifolds that
connect the moduli spaces of CICY's.
