We show that the double-copy structure of gravity forbids divergences in pure
half-maximal (16 supercharge) supergravity at four and five points at one loop
in D<8 and at two loops in D<6. We link the cancellations that render these
supergravity amplitudes finite to corresponding ones that eliminate forbidden
color factors from the divergences of pure nonsupersymmetric Yang-Mills theory.
The vanishing of the two-loop four-point divergence in D=5 half-maximal
supergravity is an example where a valid counterterm satisfying the known
symmetries exists, yet is not present. We also give explicit forms of
divergences in half-maximal supergravity at one loop in D=8 and at two loops in
D=6.Comment: 32 pages, revtex, 6 figures, v3 minor correction
