This paper presents the mechanical design of a new three-fingered robot hand for a robot designed to handle tableware. The finger mechanism has three joints and consists of a pair of fourbar linkage mechanisms, one small gas spring, and one feed screw mechanism. As the feed screw moves, the finger mechanism performs flexion and extension operations with its joints interlocked. The gas spring generates gripping force, which is adjusted at the position of the moving part moved by the feed screw. Therefore, the three-fingered robot hand can open and close synchronously, powered by a single motor in the base of the hand. The hand grips with mechanical flexibility. In addition, it can maintain its grip with no power supply. Tests show that the hand can successfully perform the movements required to grasp various kinds of tableware.