A fiber optic accelerometer based on fiber Bragg grating Fabry-Perot (FBG-FP) interferometer is presented. The sensor is a FBG-FP cavity which is formed with two weak fiber Bragg gratings (FBGs) in a single-mode fiber. The reflectivity of the two FBGs is 9.42% and 7.74% respectively, and the fiber between them is 10 meters long. An optical demodulation system was set up to analyze the reflected light of FBG-FP cavity. Acceleration signals of different frequencies and intensities were demodulated correctly and stably by the system. Based on analyzing the optical spectrum of weak FBG based FBG-FP cavity, we got the equivalent length of FBG-FP cavity. We used a path-matching Michelson interferometer (MI) to demodulate the acceleration signal. The visibility of the interference fringe we got was 41%~42% while the theory limit was 50%. This indicated that the difference of interferometer’s two arms and the equivalent length of FBG-FP cavity were matched well. Phase generated carrier (PGC) technology was used to eliminate phase fading caused by random phase shift and Faraday rotation mirrors (FRMs) were used to eliminate polarization-induced phase fading. The accelerometer used a compliant cylinder design and its’ sensitivity and frequency response were analyzed and simulated based on elastic mechanics. Experiment result showed that the accelerometer had a flat frequency response over the frequency range of 31-630Hz. The sensitivity was about 31dB (0dB=1rad/g) with fluctuation less than 1.5dB.