We report a multi-pass tunable diode laser absorption spectrometer based on the frequency-modulation spectroscopy (FMS) technique. It has the advantage of high scan speed and is immune to the etalon effect. A multi-pass Herriott-type cell was used in the spectrometer to increase the effective optical length to 17.5 m and compact the physical dimensions of the spectrometer to 60×30×30 cm³. Noise due to low-frequency fluctuation of the laser power and the 1/f noise in the rapid detection are sufficiently reduced by FMS. Interference fringes are effectively suppressed when the modulation frequency equals to integer or half-integer times of their free spectral range (FSR). An absorption line of C₂H₂ around 1.51 µm was recorded with the spectrometer to demonstrate its capabilities. The response frequency of the spectrometer is up to 100 kHz (10 µs) thanks to the high modulation frequency of FMS. The detection sensitivity of the spectrometer is about 240 ppb (3σ) at 100 kHz measurement repetition rate. The amplitude of the absorption signal is highly linear to the C₂H₂ concentration in the range of 300 ppb -100 ppm. Based on the Allan variation, the detection limit was determined to be 18 ppb with a detection time of 166 s.