  We obtain the exact non-perturbative thresholds of $R^4$ terms in IIB string
theory compactified to eight and seven dimensions. These thresholds are given
by the perturbative tree-level and one-loop results together with the
contribution of the D-instantons and of the (p,q)-string instantons. The
invariance under U-duality is made manifest by rewriting the sum as a
non-holomorphic modular function of the corresponding discrete U-duality group.
In the eight-dimensional case, the threshold is the sum of a order-1 Eisenstein
series for SL(2,Z) and a order-3/2 Eisenstein series for SL(3,Z). The
seven-dimensional result is given by the order-3/2 Eisenstein series for
SL(5,Z). We also conjecture formulae for the non-perturbative thresholds in
lower dimensional compactifications and discuss the relation with M-theory.
