  We study threefolds X in a projective space having as hyperplane section a
smooth surface with an elliptic fibration. We first give a general theorem
about the possible embeddings of such surfaces with Picard number two. More
precise results are then proved for Weierstrass fibrations, both of rank two
and higher. In particular we prove that a Weierstrass fibration of rank two
that is not a K3 surface is not hyperplane section of a locally complete
intersection threefold and we give some conditions, for many embeddings of
Weierstrass fibrations of any rank, under which every such threefold must be a
cone.
