The heavy-fermion compound CeCu_{6-x}Au_{x} has become a model system for unconventional magnetic quantum criticality. For small Au concentrations 0≤x<0.16, the compound undergoes a structural transition from orthorhombic to monoclinic crystal symmetry at a temperature T_{s} with T_{s}→0 for x≈0.15. Antiferromagnetic order sets in close to x≈0.1. To shed light on the interplay between quantum-critical magnetic and structural fluctuations we performed neutron-scattering and thermodynamic measurements on samples with 0≤x≤0.3. The resulting phase diagram shows that the antiferromagnetic and monoclinic phase coexist in a tiny Au concentration range between x≈0.1 and 0.15. The application of hydrostatic and chemical pressure allows us to clearly separate the transitions from each other and to explore a possible effect of the structural transition on the magnetic quantum-critical behavior. Our measurements demonstrate that at low temperatures the unconventional quantum criticality exclusively arises from magnetic fluctuations and is not affected by the monoclinic distortion.