  We demonstrate how duality invariance of the low energy expansion of the
four-graviton amplitude in type II string theory determines the precise
coefficients of multiloop logarithmic ultraviolet divergences of maximal
supergravity in various dimensions. This is illustrated by the explicit
moduli-dependence of terms of the form D^{2k}R^4, with k<= 3, in the effective
action. Furthermore, we show that in the supergravity limit the perturbative
contributions are swamped by an accumulation of non-perturbative effects of
zero-action instantons.
