  We compute and analyse a variety of four-derivative gravitational terms in
the effective action of six- and four-dimensional type II string ground states
with N=4 supersymmetry. In six dimensions, we compute the relevant perturbative
corrections for the type II string compactified on K3. In four dimensions we do
analogous computations for several models with (4,0) and (2,2) supersymmetry.
Such ground states are related by heterotic-type II duality or type II-type II
U-duality. Perturbative computations in one member of a dual pair give a
non-perturbative result in the other member. In particular, the exact CP-even
R^2 coupling on the (2,2) side reproduces the tree-level term plus NS 5-brane
instanton contributions on the (4,0) side. On the other hand, the exact CP-odd
coupling yields the one-loop axionic interaction a.R\wedge R together with a
similar instanton sum. In a subset of models, the expected breaking of the
SL(2,Z)_S S-duality symmetry to a \Gamma(2)_S subgroup is observed on the
non-perturbative thresholds. Moreover, we present a duality chain that provides
evidence for the existence of heterotic N=4 models in which N=8 supersymmetry
appears at strong coupling.
