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In this investigation, a laboratory-scale study to extract copper (Cu) from its oxide ore (0.425–11.2 mm particle size) was conducted using varied sulfuric acid (H2SO4) concentrations (0.05–0.5 M) as a lixiviant. Through a physicochemical and mineralogical analysis of real field ore samples from the Almalyk mine heap site (Tashkent, Uzbekistan), malachite was identified as a Cu-bearing mineral. Extraction rates were analyzed according to the ore particle size and acid concentration. The Cu extraction with the smallest particle size (in 24 h) varied between 76.7% and 94.26% at varied H2SO4 concentrations (0.05–0.5 M). Almost half (50%) of Cu was extracted from the ore within 4 and 72 h of contact time for 0.425–2 mm and 5.6–11.2 mm particle sizes, respectively, using 0.15 M H2SO4. Weeklong leaching experiments with 0.5 M H2SO4 revealed a higher copper extraction rate (≥73%) from coarse ore particles (5.6–11.2 mm). Along with the copper extraction, iron (29.6 wt%), aluminum (70.2 wt%), magnesium (85.4 wt%), and calcium (44.4 wt%) were also leached out considerably through the dissolution of silicate and carbonate gangue minerals. In this study, an 80.0–94.26% copper extraction rate with reduced acid consumption (20%) proved to be a cost-effective approach.