  Dualities of M-theory are used to determine the exact dependence on the
coupling constant of the D^6R^4 interaction of the IIA and IIB superstring
effective action. Upon lifting to eleven dimensions this determines the
coefficient of the D^6R^4 interaction in eleven-dimensional M-theory. These
results are obtained by considering the four-graviton two-loop scattering
amplitude in eleven-dimensional supergravity compactified on a circle and on a
two-torus -- extending earlier results concerning lower-derivative
interactions. The torus compactification leads to an interesting
SL(2,Z)-invariant function of the complex structure of the torus (the IIB
string coupling) that satisfies a Laplace equation with a source term on the
fundamental domain of moduli space. The structure of this equation is in accord
with general supersymmetry considerations and immediately determines tree-level
and one-loop contributions to D^6R^4 in perturbative IIB string theory that
agree with explicit string calculations, and two-loop and three-loop
contributions that have yet to be obtained in string theory. The complete
solution of the Laplace equation contains infinite series' of single
D-instanton and double D-instanton contributions, in addition to the
perturbative terms. General considerations of the higher loop diagrams of
eleven-dimensional supergravity suggest extensions of these results to
interactions of higher order in the low energy expansion.
