We demonstrate highly efficient switching of optical signals between two optical fibers controlled by a single atom. The key element of our experiment is a whispering-gallery-mode bottle microresonator, which is coupled to a single atom and interfaced by two tapered fiber couplers. This system reaches the strong coupling regime of cavity quantum electrodynamics, leading to a vacuum Rabi splitting in the excitation spectrum. We systematically investigate the switching efficiency of our system, i.e., the probability that the fiber-optical switch redirects the light into the desired output. We obtain a large redirection efficiency reaching a raw fidelity of more than 60% without postselection. Moreover, by measuring the second-order correlation functions of the output fields, we show that our switch exhibits a photon-number-dependent routing capability.