We present observations of the rapid decay of a penumbral sector associated with a strong light bridge (LB) in the active region NOAA 12680 by analyzing the scattered light-corrected Solar Dynamics Observatory/Helioseismic and Magnetic Imager data. At the beginning of penumbral decay, some dark structures gradually broke away from the umbra to which they were attached. The intensity, vertical field strength, and magnetic inclination of the dark structures are intermediate between those of the umbra and penumbra. And a strong LB formed in the umbra, which originated from the intrusion of neighboring penumbral filaments. With the formation of an LB, the total magnetic flux in the whole penumbra decreased, and a penumbral sector of the sunspot rapidly disappeared on one side of the LB. After performing a partition analysis of the penumbra, it was found that the decay rate in this region of the penumbral sector is significantly accelerated after the appearance of an outward motion of magnetic flux along the LB. The area of this penumbral sector decreased from 21-16 MSH in 4 hr. The reduction in area in this penumbral sector is accompanied by a large decay rate of the magnetic flux, 2.5 x 10(19) Mx hr(-1). These results suggest that the rapid decay of the penumbral sector is associated with the LB. The appearance of dark structures in the penumbra and the strong LB in the umbra may offer a hint that the origin of massive flux migration from the sunspot umbra may have accelerated the decay of the sunspot.