  We consider the problem of birationally modifying a morphism of complete
varieties to make it a morphism from a nonsingular variety to a normal variety.
Our main result is to give a counterexample to this problem. This example also
is a counterexample to the related conjecture of Abhyankar on ``weak
simultaneous global resolution''.
  We also give some positive results. Forinstance, a positive result of this
kind is possible if we remove the separatedness condition.
