A frequency-stabilized light source emitting at 556 nm is realized by frequency doubling a 1112 nm laser, which is phase locked to a fiber-based optical frequency comb. The 1112 nm laser is either an ytterbium (Yb)-doped distributed feedback fiber laser or a master-slave laser system that uses an external cavity diode laser as a master laser. We have achieved the continuous frequency stabilization of the light source over a 5 day period. With the light source, we have completed the second-stage magneto-optical trapping (MOT) of Yb atoms using the ${{}^{1}S}_0–{{}^{3}P}_1$ intercombination transition. The temperature of the ultracold atoms in the MOT was $40\mu \text{K}$ when measured using the time-of-flight method, and this is sufficient for loading the atoms into an optical lattice. The fiber-based frequency comb is shown to be a useful tool for controlling the laser frequency in cold-atom experiments.