Superconducting nanowire single-photon detectors are an ideal match for integrated quantum photonic circuits due to their high detection efficiency for telecom wavelength photons. Quantum optical technology also requires single-photon detection with low dark count rate and high timing accuracy. Here we present very low noise superconducting nanowire single-photon detectors based on NbTiN thin films patterned directly on top of Si3N4 waveguides. We systematically investigate a large variety of detector designs and characterize their detection noise performance. Milli-Hz dark count rates are demonstrated over the entire operating range of the nanowire detectors which also feature low timing jitter. The ultra-low dark count rate, in combination with the high detection efficiency inherent to our travelling wave detector geometry, gives rise to a measured noise equivalent power at the 10(-20) W/Hz(1/2) level.