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The syntheses and magnetic susceptibilities of a pure series of rare copper minerals from the atacamite family with general formula Zn_xCu_4-x(OH)₆Cl₂ (0 ≤ x ≤ 1) are reported. The structure of these compounds features a corner-sharing triangular kagomé lattice of antiferromagnetically coupled Cu(II) ions. We correlate the onset of magnetic ordering with the mole fraction of paramagnetic Cu(II) ions located between kagomé layers and demonstrate that the fully Zn-substituted compound shows no magnetic ordering down to 2 K, resulting in a highly spin-frustrated S = ¹/₂ lattice.