  We construct the three-loop four-point amplitude of N=8 supergravity using
the unitarity method. The amplitude is ultraviolet finite in four dimensions.
Novel cancellations, not predicted by traditional superspace power-counting
arguments, render its degree of divergence in D dimensions to be no worse than
that of N=4 super-Yang-Mills theory -- a finite theory in four dimensions.
Similar cancellations can be identified at all loop orders in certain unitarity
cuts, suggesting that N=8 supergravity may be a perturbatively finite theory of
quantum gravity.
