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We have developed a cryogenic measurement system for single-photon counting, which can be used in optical experiments requiring high time resolution in the picosecond range. The system utilizes niobium nitride superconducting nanowire single-photon detectors which are integrated in a time-correlated single-photon counting (TCSPC) setup. In this work, we describe details of the mechanical design, the electrical setup, and the cryogenic optical components. The performance of the complete system in TCSPC mode is tentatively benchmarked using 140 fs long laser pulses at a repetition frequency of 75 MHz. Due to the high temporal stability of these pulses, the measured time resolution of 35 ps (FWHM) is limited by the timing jitter of the measurement system. The result was cross-checked in a Coherent Anti-stokes Raman Scattering (CARS) setup, where scattered pulses from a β-barium borate crystal have been detected with the same time resolution.