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Stellar winds from massive stars may be significant sources of cosmic rays (CRs). To investigate this connection, we report a detailed study of gamma-ray emission near the young Milky Way star cluster (≈0.5 Myr old) in the star-forming region RCW 38 and compare this emission to its stellar wind properties and diffuse X-ray emission. Using 15 yr of Fermi-LAT data in the 0.2–300 GeV band, we find a significant ( σ > 22) detection coincident with the star cluster, producing a total gamma-ray luminosity (extrapolated over 0.1–500 GeV) of L _γ =(2.66 ± 0.92) × 10 ^34 erg s ^−1 adopting a power-law spectral model (Γ = 2.34 ± 0.04). Using an empirical relationship and STARBURST99 , we estimate the total wind power to be 8 × 10 ^36 erg s ^−1 , corresponding to a CR acceleration efficiency of η _CR ≃ 0.4 for an assumed diffusion coefficient consistent with D = 10 ^28 cm ^2 s ^−1 . Alternatively, a lower acceleration efficiency of 0.1 can produce this L _γ if the diffusion coefficient is smaller, D ≃ 2.5 × 10 ^27 cm ^2 s ^−1 . Additionally, we analyze Chandra X-ray data from the region and compare the hot-gas pressure to the CR pressure. We find the former is 4 orders of magnitude greater, suggesting that the CR pressure is not dynamically important relative to stellar winds. As RCW 38 is too young for supernovae to have occurred, the high CR acceleration efficiency in RCW 38 demonstrates that stellar winds may be an important source of Galactic CRs.