A distributed vibration sensing system with a high signal-to-noise ratio (SNR) based on an ultra-weak fiber Bragg grating is proposed in this paper. The system uses an online inscribed chirped grating array with a 3 dB bandwidth greater than 3.4 nm and a reflectivity of 10 ^-5 as the sensing fiber, and phase demodulation is realized through typical dual-pulse modulation and a 3×3 coupler. To eliminate interference noises from optical components including laser, modulator, and optical amplifiers, we employ the least mean square algorithm and a symmetric detection structure for adaptive filtering. The experimental results show that compared with the wavelet denoising method, the method can greatly decrease the system noise. It can reduce the high-frequency noise floor of the 1 kHz demodulated signal by 40.1 dB at the maximum, and the SNR is as high as 96.5 dB. For the 2 km long sensing fiber, the strain sensitivity of the system reaches 13.66 pε/√Hz at 1-5 kHz.