This paper presents the design and control of a novel gripper device with a passive type of compliant constant-force mechanism. The function of constant-force output is achieved by using the combination of a positive-stiffness and a negative-stiffness mechanism. The negative stiffness is generated by a bistable buckled fixed-guided beam. Conventionally, when gripping a target-object using a high-stiffness gripper, both the displacement control and force control are required to prevent the damage of the object. The uniqueness of the developed constant-force gripper is that it can eliminate the dependence on force control while keeping the force constant via its mechanical structure. The gripper performance is evaluated by the established analytical model and nonlinear finite-element analysis, and validated through experimental study on a fabricated prototype. To achieve a precise position output of the gripper jaw, a discrete-time variable structure control strategy based on a nonswitching type of reaching law is realized. The effectiveness of the gripper system is verified by conducting experimental studies on grasp-hold-release operations of a micro copper wire.