A covariant path-integral calculation of the even-spin structure contribution to the one-loop N-graviton scattering amplitude in the type-II superstring theory is presented. The apparent divergence of the N = 5 amplitude is resolved by separating it into twelve independent terms corresponding to different orders of inserting the graviton vertex operators. Each term is well defined in an appropriate kinematic region and can be analytically continued to physical regions where it develops branch cuts required by unitarity. The zero-slope limit of the N = 5 amplitude is performed, and the Feynman diagram content of the low-energy field theory is examined. Both one-particle irreducible (1PI) and one-particle reducible (1PR) graphs with massless internal states are generated in this limit. One set of 1PI graphs has the same divergent dependence on the cut-off as that found in the four-graviton case, and it is proved that such graphs exist for all N. The 1PR graphs are contributed by the poles in the world-sheet chiral Green functions.