The measuring accuracy of the fiber optic gyroscope (FOG) for weak signals under very short sampling time is significantly impacted by the quantization error, impeding its application in high-speed measurement and real-time control. In this work, we propose and implement a double-electrode-pair multifunction integrated-optic circuit (MIOC), which contains an additional pair of short electrodes besides the conventional electrode-pair. Taking advantage of the better modulating precision of the additional electrode-pair, the digital feedback is more refined and the quantization error in the FOG output is significantly suppressed. The driving circuits and the control scheme of the proposed MIOC are specially designed for FOGs. Experimental results show that the resolution for extremely small angular rates at short smoothing times is significantly improved. This work provides the potential of the applications in high-speed measuring and controlling systems for high-precision FOGs.