We have fabricated and tested single photon detectors based on a current biased superconducting niobium nanowire patterned into a meander. The detectors are fabricated from high quality, ultra high vacuum sputtered niobium thin films on a sapphire substrate. For detection of single optical photons, we show that the superconductor's intrinsic kinetic inductance does not limit the reset time of the detector, which is ≈ 2 nanoseconds, in contrast to the longer reset times seen in niobium nitride detectors of similar size and geometry. We also describe a readout scheme for photon counting that is unique to Nb due to its lower resistivity. These detectors have applications in imaging of infrared photoemission from CMOS logic circuits as well as in optical communications and quantum information processing.