  We consider the one-loop B^2h^3 and B^4h amplitudes in type II string theory,
where B is the NS-NS two-form and h the graviton, and expand to lowest order in
alpha'. After subtracting diagrams due to quartic terms in the effective
action, we determine the presence and structure of both an H^2R^3 and
H^2(DH)^2R term. We show that both terms are multiplied by the usual
(t_8t_8\pm{1/8}\epsilon_{10}\epsilon_{10}) factor.
