We report on high-pressure angle-dispersive synchrotron X-ray diffraction data of a natural Zn3.78(2)Cu1.22(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 κa0 = 3.79(5)·10−3 GPa−1, κb0 = 5.44(9)·10−3 GPa−1 and κc0 = 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.