  We prepare and detect the hyperfine state of a single 87Rb atom coupled to a
fiber-based high finesse cavity on an atom chip. The atom is extracted from a
Bose-Einstein condensate and trapped at the maximum of the cavity field,
resulting in a reproducibly strong atom-cavity coupling. We use the cavity
reflection and transmission signal to infer the atomic hyperfine state with a
fidelity exceeding 99.92% in a read-out time of 100 microseconds. The atom is
still trapped after detection.
