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We propose a brane-world setup based on gauge/gravity duality that permits the simultaneous realisation of self-tuning of the cosmological constant and a stabilisation of the electroweak hierarchy. The Standard Model dynamics including the Higgs sector is confined to a flat 4-dimensional brane, embedded in a 5-dimensional bulk whose dynamics is governed by Einstein-dilaton-axion gravity. The inclusion of a dynamical bulk axion is new compared to previous implementations of the self-tuning mechanism. Because of the presence of the axion, the model generically exhibits a multitude of static solutions, with different values for the equilibrium position for the brane. Under mild assumptions regarding the dependence of brane parameters on bulk fields, a number of these solutions exhibit electroweak symmetry breaking with a hierarchically small Higgs mass as compared to the cutoff-scale of the brane theory. The realisation of self-tuning of the cosmological constant is generic and as efficient as in previous constructions without a bulk axion. Vacua with a hierarchically small Higgs mass can sometimes be found, regardless of whether the brane theory depends explicitly on the bulk axion. Because it is expected on general principles that the brane action will depend on the axion, the generation of solutions with a large hierarchy is a robust feature.