  In hep-th/9903210 (curvature)$^2$ terms of the effective D-brane action were
derived to lowest order in the string coupling. Their results are correct up to
ambiguous terms which involve the second fundamental form of the D-brane. We
compute five point string amplitudes on the disk. We compare the subleading
order in $\alpha'$ of the string amplitudes with the proposed lagrangian of
hep-th/9903210 supplemented by the ambiguous terms. The comparison determines
the complete form of the gravitational terms in the effective D-brane action to
order ${\calO}(\alpha^{' 2})$. Our results are valid for arbitrary ambient
geometries and world-volume embeddings.
