An unusual spin excitation mode observed by neutron scattering has inspired numerous theoretical studies of the interplay between charged quasiparticles and collective spin excitations in the copper oxide high-temperature superconductors. The mode has, thus far, only been observed in materials with crystal structures consisting of copper oxide bilayers, and it is absent in the single-layer compound La(2-x)Sr(x)CuO(4+delta). Neutron-scattering data now show that the mode is present in Tl(2)Ba(2)CuO(6+delta), a single-layer compound with a superconducting transition temperature of approximately 90 kelvin, demonstrating that it is a generic feature of the copper oxide superconductors, independent of the layer sequence. This restricts the theoretical models for the origin of the resonant mode and its role in the mechanism of high-temperature superconductivity.