  Evanescent operators such as the Gauss-Bonnet term have vanishing
perturbative matrix elements in exactly D=4 dimensions. Similarly, evanescent
fields do not propagate in D=4; a three-form field is in this class, since it
is dual to a cosmological-constant contribution. In this Letter, we show that
evanescent operators and fields modify the leading ultraviolet divergence in
pure gravity. To analyze the divergence, we compute the two-loop
identical-helicity four-graviton amplitude and determine the coefficient of the
associated (non-evanescent) R^3 counterterm studied long ago by Goroff and
Sagnotti. We compare two pairs of theories that are dual in D=4: gravity
coupled to nothing or to three-form matter, and gravity coupled to zero-form or
to two-form matter. Duff and van Nieuwenhuizen showed that, curiously, the
one-loop conformal anomaly --- the coefficient of the Gauss-Bonnet operator ---
changes under p-form duality transformations. We concur, and also find that the
leading R^3 divergence changes under duality transformations. Nevertheless, in
both cases the physical renormalized two-loop identical-helicity four-graviton
amplitude can be chosen to respect duality. In particular, its
renormalization-scale dependence is unaltered.
