Based on the interferometric demodulation technique, a fiber Bragg grating sensor system employing an unbalance wavelength scanning Michelson interferometer to transpose the strain related spectral signal into phase signal is reported. The strain can be recovered by measuring the phase change of the signal with a phase-meter. Experiment demonstrated that the sensing sensitivity of this system with a resolution of 5.5 nε is 1.80° /με. It basically matches the prospected value of 1.83° /με.