The dark-count rate (DCR) is a key parameter of single-photon detectors. By introducing a bulk optical band-pass filter mounted on a fiber-to-fiber optical bench cooled at 3 K and blocking down to 5 μm, we suppressed the DCR of a superconducting nanowire single-photon detector by more than three orders of magnitude. The DCR is limited by the blackbody radiation through a signal passband of 20-nm bandwidth. The figure of merit, system detection efficiency, and DCR were 2.7×10(11), 2.3%, and 10(-3) Hz, respectively. Narrowing the bandwidth to 100 GHz suppresses the DCR to 10(-4) Hz, and the figure of merit increases to 1.8×10(12).