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The neutron-capture reaction plays a critical role in the synthesis of the elements in stars and is important for societal applications including nuclear power generation and stockpile-stewardship science. However, it is difficult-if not impossible-to directly measure neutron capture cross sections for the exotic, short-lived nuclei that participate in these processes. In this Letter we demonstrate a new technique which can be used to indirectly determine neutron-capture cross sections for exotic systems. This technique makes use of the (d,p) transfer reaction, which has long been used as a tool to study the structure of nuclei. Recent advances in reaction theory, together with data collected using this reaction, enable the determination of neutron-capture cross sections for short-lived nuclei. A benchmark study of the ^{95}Mo(d,p) reaction is presented, which illustrates the approach and provides guidance for future applications of the method with short-lived isotopes produced at rare isotope accelerators.