Within the adiabatic approximation, time-dependent density functional theory yields only single excitations. Near states of double excitation character, the exact exchange-correlation kernel has a strong dependence on frequency. We derive the exact frequency-dependent kernel when a double excitation mixes with a single excitation, well separated from the other excitations, in the limit that the electron-electron interaction is weak. Building on this, we construct a nonempirical approximation for the general case, and illustrate our results on a simple model. (C) 2004 American Institute of Physics.