The prospect of quantum networks, in which quantum information is carried by single photons in photonic circuits, has long been the driving force behind the effort to achieve all-optical routing of single photons. We realized a single-photon-activated switch capable of routing a photon from any of its two inputs to any of its two outputs. Our device is based on a single atom coupled to a fiber-coupled, chip-based microresonator. A single reflected control photon toggles the switch from high reflection (R similar to 65%) to high transmission (T similar to 90%), with an average of similar to 1.5 control photons per switching event (similar to 3, including linear losses). No additional control fields are required. The control and target photons are both in-fiber and practically identical, making this scheme compatible with scalable architectures for quantum information processing.