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We demonstrate detection of short pulses of THz radiation with a fast and sensitive antenna-coupled NbN hot-electron bolometer (HEB). The bolometer offers an optical noiseequivalent power of 2.0×10-13 W·Hz-0.5 and a time constant of 35 ps. It combines the sensitivity similar to that of the state-of-the-art THz detectors operating at liquid helium with a picosecond response time. We employed difference-frequency generation in a GaSe crystal driven by a Nd:YLF laser operating on 1.047-μm and 1.053-µm lines to obtain short pulses of THz radiation at 1.64 THz. Using the HEB as a detector has led to a better signal-to-noise ratio and has allowed representing the transients of THz pulses without distorted power and timing characteristics. The compact and easy to use source of THz radiation, together with a fast and sensitive THz detector based on the HEB, is a promising candidate for using in a number of important applications such as THz scanning systems or scattering near-field microscopy.