Hard X-ray spectroscopy is an essential diagnostics used to understand physical processes that take place in high energy density plasmas produced by intense laser-plasma interactions. A bundle of hard X-ray detectors, of which the responses have different energy thresholds, is used as a conventional single-shot spectrometer for high-flux (>10(13) photons/shot) hard X-rays. However, high energy resolution (Δhv/hv < 0.1) is not achievable with a differential energy threshold (DET) X-ray spectrometer because its energy resolution is limited by energy differences between the response thresholds. Experimental demonstration of a Compton X-ray spectrometer has already been performed for obtaining higher energy resolution than that of DET spectrometers. In this paper, we describe design details of the Compton X-ray spectrometer, especially dependence of energy resolution and absolute response on photon-electron converter design and its background reduction scheme, and also its application to the laser-plasma interaction experiment. The developed spectrometer was used for spectroscopy of bremsstrahlung X-rays generated by intense laser-plasma interactions using a 200 μm thickness SiO2 converter. The X-ray spectrum obtained with the Compton X-ray spectrometer is consistent with that obtained with a DET X-ray spectrometer, furthermore higher certainly of a spectral intensity is obtained with the Compton X-ray spectrometer than that with the DET X-ray spectrometer in the photon energy range above 5 MeV.