Force regulation is a challenging issue of robot end-effectors when interacting with unknown environments. It often requires sophisticated sensors with computerized control. This paper presents a constant-force mechanism (CFM) to regulate the contact force of a robot end-effector. The proposed CFM is a monolithic compliant mechanism that has no frictional wear and is capable of miniaturization. Due to the passive mechanism, additional sensors and control effort are minimized. We propose a design formulation to find the optimal CFM shape that produces the most constant force. The reaction force to input displacement curve is invariant of size and flexural rigidity. The curve can be manipulated depending on the desirable situations. The CFM is validated through an experiment. When equipped with the CFM, an illustrative end-effector can adapt to a surface of variable height, without additional motion programming. With the merits shown, we expect this type of elastic mechanism can be utilized in robot end-effectors to provide friendly contact with environment.