Abstract: A driving force transmission device includes a worm meshing with a worm wheel. The worm has an outer casing defining a first helical groove on the outer peripheral surface thereof, and an inner cylinder mounted in the outer casing and provided on the outer peripheral surface thereof with a helical groove. The outer casing has ball transfer holes at both ends of the first groove and which transfer holes communicate with the helical groove in the inner cylinder. A plurality of balls are rollably and loosely fitted in the helical groove in the outer periphery of the inner cylinder, the first groove in the outer periphery of the outer casing and the second groove in the outer periphery of the worm wheel. The balls circulate between the helical groove in the outer periphery of the inner cylinder and the first helical groove in the outer periphery of the outer casing via the ball transfer holes. Frame rings mounted on the worm at both ends thereof prevent the balls from floating out of the first groove.