In this paper, we report inelastic neutron scattering measurements of the magnetic excitations of green dioptase Cu6 [Si6 O18] ∙ 6H2O. The observed spectrum contains two magnetic modes and a prominent spin gap that is consistent with the ordered ground state of Cu moments coupled antiferromagnetically in spiral chains along the c axis and ferromagnetically in ab planes on the hexagonal cell. The data are in excellent agreement with a spin -½ Hamiltonian that includes antiferromagnetic nearest-neighbor intrachain coupling Jc = 10.6 (1) meV, ferromagnetic interchain coupling Jab = -1.2 (1) meV, and exchange anisotropy ΔJc = 0.14 (1) meV. We calculated the sublattice magnetization to be strongly reduced, ~ 0.39µB. This appears compatible with a reduced Néel temperature, TN = 14.5 K ≪ Jc, and can be explained by a presence of quantum spin fluctuations.