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We demonstrate cooling of 10⁴ antiprotons in a dense, cold plasma of ∼10⁸ positrons, confined in a nested cylindrical Penning trap at about 15 K. The time evolution of the cooling process has been studied in detail, and several distinct types of behavior identified. We propose explanations for these observations and discuss the consequences for antihydrogen production. We contrast these results with observations of interactions between antiprotons and “hot” positrons at about 3000 K, where antihydrogen production is strongly suppressed.