We propose a new neutron-capture site in early metal-poor and metal-free stars of ∼20-30 M o that results from proton ingestion in the He shell during late stages of the stars' lives. Most of the neutron capture occurs in the first ≲106 s following proton ingestion when 13C(α, n)16O produces neutron densities typical of the intermediate neutron-capture process. This phase may be followed by another lasting 107 s with 17O(α, n)20Ne producing much lower neutron densities typical of the slow neutron-capture process. We explore the dependence of the proposed neutron-capture nucleosynthesis on the amount and time of proton ingestion, the initial metallicity, and the ensuing supernova shock. We obtain a range of heavy-element abundance patterns, including those attributed to the slow neutron-capture process or a combination of the slow and rapid neutron-capture processes. Our results can account for the observed ubiquity of heavy elements such as Sr and Ba in the early Galaxy and explain puzzling abundance patterns of these elements in at least some very metal-poor (VMP) stars, including those of the carbon-enhanced varieties. In the latter case, the explanation by the single site proposed here differs from the existing paradigm that attributes various classes of VMP stars to enrichment by multiple different sites. © 2018. The American Astronomical Society. All rights reserved.