The high-field magnetic properties and magnetic order of the gem mineral green dioptase Cu6[Si6O18]·6H2O have been studied by means of single-crystal neutron diffraction in magnetic fields up to 21 T and magnetization measurements up to 30 T. In zero field, the Cu2+ moments in the antiferromagnetic chains are oriented along the c axis with a small off-axis tilt. For a field applied parallel to the c axis, the magnetization shows a spin-flop-like transition at B∗=12.2 T at 1.5 K. Neutron diffraction experiments show a smooth behavior in the intensities of the magnetic reflections without any change in the periodicity of the magnetic structure. Bulk and microscopic observations are well described by a model of ferromagnetically coupled antiferromagnetic XXZ spin-12 chains, taking into account a change of the local easy-axis direction. We demonstrate that the magnetic structure evolves smoothly from a deformed Néel state at low fields to a deformed spin-flop state in a high field via a strong crossover around B∗. The results are generalized for different values of interchain coupling and spin anisotropy.