  We present a global conformal invariant on closed six-manifolds which
obstructs the existence of a conformally Einstein metric. We show that this
obstruction is nontrivial and, up to multiplication by a constant, is the
unique such invariant. This also gives rise to a (possibly trivial)
diffeomorphism invariant which obstructs the existence of an Einstein metric.
We also discuss global conformal invariants which obstruct the existence of a
conformally Einstein metric on closed six-manifolds with infinite fundamental
group.
