  Four-graviton couplings in the low energy effective action of type II string
vacua compactified on tori are strongly constrained by supersymmetry and
U-duality. While the $R^4$ and $D^4 R^4$ couplings are known exactly in terms
of Langlands-Eisenstein series of the U-duality group, the $D^6 R^4$ couplings
are not nearly as well understood. Exploiting the coincidence of the U-duality
group in $D=6$ with the T-duality group in $D=5$, we propose an exact formula
for the $D^6 R^4$ couplings in type II string theory compactified on $T^4$, in
terms of a genus-two modular integral plus a suitable Eisenstein series. The
same modular integral computes the two-loop correction to $D^6 R^4$ in 5
dimensions, but here provides the non-perturbative completion of the known
perturbative terms in $D=6$. This proposal hinges on a systematic re-analysis
of the weak coupling and large radius of the $D^6 R^4$ in all dimensions $D\geq
3$, which fills in some gaps and resolves some inconsistencies in earlier
studies.
