A novel demodulation method for weak signal of fiber sagnac interferometer is proposed, by which we obtained a simple long distance optical fiber sensor system with positioning accuracy less than 1 km. In our work, we expanded one channel of interference signal in a Taylor series, and the weak phase difference of interference signal could be obtained by a weak signal approximation theory from the Taylor series, then the approximate phase difference was used to demodulate the weak signal and locate the fiber vibration. By using the proposed method, there was no need to use phase amplification technology such as the phase generated carrier modulation or phase-locked amplification technology for weak signal demodulation, whereas, we just need one channel to demodulate the phase, therefore it simplified the system structure and demodulation algorithm, and cut down the cost. Based on this new demodulation theory, the strong signal could also be demodulated; it just need choose a weak part from the strong signal. This proposed method was demonstrated in a long-distance fiber sagnac interferometer sensor system with length of 50 km. The positioning accuracy was less than 1 km in the length of 10, 20, 30 and 40 km. We believe the accuracy could be enhanced with the help of signal processing technology such as signal filter and signal denoising.