  Birational Calabi-Yau threefolds in the same deformation family provide a
`weak' counterexample to the global Torelli problem, as long as they are not
isomorphic. In this paper, it is shown that deformations of certain
desingularized weighted complete intersections provide examples of families
containing birational varieties. The constructed examples are shown to be
non-isomorphic using a specialization argument.
