We address the problem of learning feedback control where the controller is a network constructed solely of deterministic spiking neurons. In contrast to previous investigations that were based on a spike rate model of the neuron, the control signal here is determined by the precise temporal positions of spikes generated by the output neurons of the network. We model the problem formally as a hybrid dynamical system comprised of a closed loop between a plant and a spiking neuron network. We derive a novel synaptic weight update rule via which the spiking neuron network controller learns to hold process variables at desired set points. The controller achieves its learning objective based solely on access to the plant's process variables and their derivatives with respect to changing control signals; in particular, it requires no internal model of the plant. We demonstrate the efficacy of the rule by applying it to the classical control problem of the cart-pole (inverted pendulum) and a model of fish locomotion. Experiments show that the proposed controller has a stability region comparable to a traditional PID controller, its trajectories differ qualitatively from those of a PID controller, and in many instances the controller achieves its objective using very sparse spike train outputs.