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Abstract We present a survey undertaken with the Atacama Large Millimeter/submillimeter Array (ALMA) to study the galaxies associated with a representative sample of 16 damped Lyαabsorbers (DLAs) atz ≈ 4.1–4.5, using the [Cii] 158μm ([Cii]) line. We detect seven [Cii]-emitting galaxies in the fields of five DLAs, all of which have absorption metallicity [M/H] > −1.5. We find that the detectability of these Hi-selected galaxies with ALMA is a strong function of DLA metallicity, with a detection rate of $71_{-20}^{+11}$% for DLAs with [M/H] > −1.5 and 0⁺¹⁸% for DLAs with [M/H] < −1.5. The identified DLA galaxies have far-IR properties similar to those of typical star-forming galaxies atz ∼ 4, with estimated obscured star formation rates ranging from ≲6M_⊙yr⁻¹to 110M_⊙yr⁻¹. High-metallicity DLAs therefore provide an efficient way to identify and study samples of high-redshift, star-forming galaxies, without preselecting the galaxies by their emission properties. The agreement between the velocities of the metal absorption lines of the DLA and the [Cii] emission line of the DLA galaxy indicates that the metals within the DLA originated in the galaxy. With observed impact parameters between 14 and 59 kpc, this indicates that star-forming galaxies atz ∼ 4 have a substantial reservoir of dense, cold, neutral gas within their circumgalactic medium that has been enriched with metals from the galaxy.