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This paper presents a lightweight (328 g) anthropomorphic robot hand that can exert a large grasping force. We propose a mechanism combining flexion drives and a force-magnification drive for a cable-driven multifingered robot hand. The flexion drive consisting of a feed screw enables quick finger motion, and the force-magnification drive, consisting of an eccentric cam, a bearing, and a pulley, enables a firm grasp. This paper also proposes a 3-D linkage for the thumb. This linkage consists of four links and is driven by a feed screw. It can oppose to a large force exerted by the force-magnification drive. These mechanisms are compact enough to be installed in the developed lightweight hand. We experimentally verify that the maximum fingertip force of the hand exceeds 20 N and that the thumb can hold a large force of 100 N. The time to fully close the hand using the flexion drives is 0.47 s. After the fingers make contact with an object, the time to achieve a firm grasp using the grasp-magnification drive is approximately 1 s.