It was observed that repetitive and task-oriented movements can strengthen muscles and improve the walking capabilities among patients experiencing gait impairments due to neurological disorders. However, the traditional physiotherapy is laborious, may not provide desired cadence and gait patterns, and due to the lack of therapists, it remains inaccessible to many patients. In the present work, we propose a novel low-cost treadmill-based underactuated gait rehabilitation exoskeleton that can provide naturalistic lower limb movements during walking. The exoskeleton mechanism has been devised by constraining a planar serial chain formed by revolute joints to resemble the Stephenson III six-bar linkage satisfying the adaptability and the end-effector trajectory requirements. The synthesized mechanism matches the concurrent knee and ankle joint movements during human walking relative to the hip, in terms of position and time. The kinematics and dynamic force analysis of the mechanism formulating the position, velocity, acceleration, and static torque have been presented. A full working lightweight exoskeleton prototype with a single actuated joint has been constructed. A pilot study with a healthy human subject has been performed. As a result, it is found that the proposed exoskeleton can provide naturalistic human motion trajectories suitable for rehabilitation purposes.