Force regulation is a challenging problem for robot end-effectors when interacting with an unknown environment. It often requires sophisticated sensors with computerized control. This paper presents an adjustable constant-force mechanism (ACFM) to passively regulate the contact force of a robot end-effector. The proposed ACFM combines the negative stiffness of a bistable mechanism and positive stiffness of a linear spring to generate a constant-force output. Through prestressing the linear spring, the constant-force magnitude can be adjusted to adapt to different working environments. The ACFM is a monolithic compliant mechanism that has no frictional wear and is capable of miniaturization. We propose a design formulation to find optimal mechanism configurations that produce the most constant-force. A resulting force to displacement curve and maximal stress curve can be easily manipulated to fit a different application requirement. Illustrated experiments show that an end-effector equipped with the ACFM can adapt to a surface of variable height, without additional motion programming. Since sensors and control effort are minimized, we expect this mechanism can provide a reliable alternative for robot end-effectors to interact friendly with an environment.