This paper is concerned with stellar neutron capture processes which occur at a rate slow compared to the intervening beta decays, the so-called s-process in the synthesis of the elements. An approximate method of high reliability has been devised to solve for the abundance distributions resulting from the exposure of seed nuclei, such as Fe 56, to a weak neutron flux in stars. The capture chain differential equations are solved by approximately matching the Laplace transforms of the exact solutions to the Laplace transform of an easily calculable function. From the sequence of abundance distributions generated in this manner for specified numbers of neutrons per initial seed nucleus, one can estimate the superpositions of neutron exposures required to reproduce the experimentally observed abundance distribution for the s-process isotopes of the elemments. Not only can the validity of the s-process model of heavy element synthesis in stellar interiors be demonstrated in this way, but certain inferences about the "history" of stellar neutron processes also appear. Special attention is paid in this regard to the "terminal" exposures which have synthesized lead and bismuth at the end of the line in the s-process. An analysis is appended of neutron capture cross sections near 25 kev for the s-process nuclei, including interpolations based upon empirical cross sections guided where necessary by isotopic and elemental abundances. A complete correlation between neutron capture cross sections and s-process abundances cannot be made at the present stage of knowledge, but the methods described will lead to an eventual solution as more empirical information becomes available.