We report on high-pressure angle-dispersive synchrotron X-ray diffraction data of a natural Zn_3.78(2)Cu_1.22(2)(CO₃)₂(OH)₆ aurichalcite mineral up to 7.6 GPa and ab initio total energy calculations of the aurichalcite structure with three different Zn-Cu stoichiometries (Zn:Cu ratios = 10:0, 8:2 and 6:4). A monoclinic-to-triclinic displacive second-order phase transition was found experimentally at 3 GPa. The experimental bulk modulus of the initial P2₁/m aurichalcite is B₀ = 66(2) GPa, with a first-pressure derivative of B₀′ = 9(2). A comparison with other basic copper and zinc carbonates shows that this B₀ value is considerably larger than those of malachite and azurite. This relative incompressibility occurs despite the fact that aurichalcite features a layered structure due to the number of directed hydrogen bonds between carbonate groups and the cation-centered oxygen polyhedra forming complex sheets. The existence of different bond types and polyhedral compressibilities entails a certain anisotropic compression, with axial compressibilities κ_a0 = 3.79(5)·10⁻³ GPa⁻¹, κ_b0 = 5.44(9)·10⁻³ GPa⁻¹ and κ_c0 = 4.61(9)·10⁻³ GPa⁻¹. Additional density-functional theory calculations on the C2/m hydrozincite-type structure with different Zn:Cu compositional ratios shows that the aurichalcite structure is energetically more stable than the hydrozincite one for compositions of Zn:Cu = 10:0, 8:2 and 6:4 at room pressure. The pure Zn aurichalcite phase, however, was predicted to transform into hydrozincite at 18 GPa, which suggests that the experimentally observed hydrozincite structure is a metastable phase.