A Compton-scattering-based X-ray spectrometer is developed to obtain the energy distribution of fast electrons produced by intense laser and matter interactions. Bremsstrahlung X-rays generated by fast electrons in a material are used to measure fast electrons' energy distribution in matter. In the Compton X-ray spectrometer, X-rays are converted into recoil electrons by Compton scattering in a converter made from fused silica glass, and a magnet-based electron energy analyzer is used to measure the energy distribution of the electrons that recoil in the direction of the incident X-rays. The spectrum of the incident X-rays is reconstructed from the energy distribution of the recoil electrons. The accuracy of this spectrometer is evaluated using a quasi-monoenergetic 6 MeV electron bunch that emanates from a linear accelerator. An electron bunch is injected into a 1.5 mm thick tungsten plate to produce bremsstrahlung X-rays. The spectrum of these bremsstrahlung X-rays is obtained in the range from 1 to 9 MeV. The energy of the electrons in the bunch is estimated using a Monte Carlo simulation of particle-matter interactions. The result shows that the spectrometer's energy accuracy is ±0.5 MeV for 6.0 MeV electrons.