  We introduce lossless state detection of trapped neutral atoms based on
cavity-enhanced fluorescence. In an experiment with a single 87-Rb atom, a
hyperfine-state-detection fidelity of 99.4% is achieved in 85 microseconds. The
quantum bit is interrogated many hundreds of times without loss of the atom
while a result is obtained in every readout attempt. The fidelity proves robust
against atomic frequency shifts induced by the trapping potential. Our scheme
does not require strong coupling between the atom and cavity and can be
generalized to other systems with an optically accessible quantum bit.
