Ultraviolet properties of Maximal Supergravity

In a recent paper we suggested that the dualities of M-theory might imply that the four-graviton amplitude of four-dimensional N=8 supergravity is ultraviolet finite in four dimensions. In this paper we argue that even without invoking duality conjectures, but by making direct use of string perturbation theory as a covariant ultraviolet regulator, the four-graviton amplitude of four-dimensional N=8 supergravity should be ultraviolet finite up to at least eight loops. We further argue that the h-loop M-graviton amplitude should be finite for h