  Using a single neutral \textsuperscript{87}Rb atom held in an optical trap,
and "quantum jump" detection of single-photon-initiated state changes, we
demonstrate an intrinsically-narrowband single-photon detector, of interest for
separating weak signals from strong optical background. Using novel statistical
analysis, we measure quantum efficiency of \SI{2.9+-0.2e-3}{}, a record for
single-pass quantum jump production, and dark counts of
\SI{9+-20e-3}{counts\per\second} during passive accumulation plus
\SI{1.8+-0.1e-2}{counts} per readout, orders of magnitude below those of
traditional single-photon detectors. The \SI{6}{\mega\hertz} detection
bandwidth is orders of magnitude narrower than existing atomic filters.
Available methods can substantially improve \QJPDAcronym{} quantum efficiency,
dark counts, bandwidth, and tunability.
