A welding apparatus and a welding method are employed for laser narrow groove welding which performs welding scanning a laser beam in the welding direction while feeding a solid filler metal into a narrow groove. The welding apparatus includes a laser beam irradiation head having a mechanism periodically oscillating an irradiation point of the laser beam with a predetermined amplitude in the bottom of the groove, and a filler metal control device having a solid filler metal feeder feeding the solid filler metal to the molten pool formed in the bottom of the groove by the laser beam and adjusting the feeding position independent of a motion of the laser beam irradiation head so that the tip position of the solid filler metal detected is constantly positioned in the center of the groove.