This brief introduces a simple circuit solution to secure loop stability of an analog-domain fast-frequency offset cancellation loop (OCL). The OCL is composed of a low-IF receiver, phase-domain frequency offset detector (OD), and fractional-N phase-locked loop (PLL). Since the OCL uses a phase-domain OD, a stability concern is essentially needed for its practical use. From the frequency-domain analysis, a PLL bandwidth adaptation by controlling charge-pump currents is proposed to achieve a strong stability with phase-margin of more than 60 degrees. Additionally, a tradeoff between the OCL accuracy and hardware complexity is discussed, and a 'design example is shown for the 2.4-GHz ZigBee application. With 4-MHz IF, designed for an 0.18-mu m CMOS process, our circuit takes 30 mu s to reject the frequency offset of +200 kHz within the accuracy of +/- 5 ppm, with 60-DFFs for a time-to-digital converter.