This paper proposes the design of a wholly mechanical constant-force gripper that can accommodate the imprecise manipulation of brittle/delicate objects by the actuation. This was achieved by designing a constant-force mechanism as the jaw that allowed a constant force to be applied to the grasping objects regardless of the displacement of the mechanism. The constant-force mechanism is attached to the end effector of the gripper via a parallelogram mechanism which ensures that the jaws remain in parallel. The constant-force mechanism combines the negative stiffness of a bistable mechanism and the positive stiffness of a linear spring to generate a constant force output. By preloading the positive stiffness mechanism, the magnitude of the constant force can be adjusted to be as low as zero. The constant-force mechanism has been fully modelled and simulated using finite element analysis. A normalised force-displacement curve has been developed that allows to obtain the simplified analytical negative stiffness of the bistable mechanism. The design formulation to find the optimal configuration that produces the most constant force has been developed. Illustrated experiments prove the concept of the design although the discrepancies between finite element analysis results and testing results exist due to bistable beam manufacturing error.