  The R^4 terms in the effective action for M-theory compactified on a
two-torus are motivated by combining one-loop results in type II superstring
theories with the Sl(2,Z) duality symmetry. The conjectured expression
reproduces precisely the tree-level and one-loop R^4 terms in the effective
action of the type II string theories compactified on a circle, together with
the expected infinite sum of instanton corrections. This conjecture implies
that the R^4 terms in ten-dimensional string type II theories receive no
perturbative corrections beyond one loop and there are also no non-perturbative
corrections in the ten-dimensional IIA theory. Furthermore, the
eleven-dimensional M-theory limit exists, in which there is an R^4 term that
originates entirely from the one-loop contribution in the type IIA theory and
is related by supersymmetry to the eleven-form C^{(3)}R^4. The generalization
to compactification on T^3 as well as implications for non-renormalization
theorems in D-string and D-particle interactions are briefly discussed.
