  Single photons from a coherent input are efficiently redirected to a separate
output by way of a fiber-coupled microtoroidal cavity interacting with
individual Cesium atoms. By operating in an overcoupled regime for the
input-output to a tapered fiber, our system functions as a quantum router with
high efficiency for photon sorting. Single photons are reflected and excess
photons transmitted, as confirmed by observations of photon antibunching
(bunching) for the reflected (transmitted) light. Our photon router is robust
against large variations of atomic position and input power, with the observed
photon antibunching persisting for intracavity photon number 0.03 \lesssim n
\lesssim 0.7.
