In this letter, we present an underactuated soft exoskeleton which can sequentially provide knee extension and ankle plantarflexion assistance for each leg with only one motor. The aim of this work is to assist the motions that have chronological moments at lower-limb joints and minimize the mass carried by the wearer. The underactuated soft exoskeleton consists of a novel actuation mechanism: Continuum-Cable System (CCS), which allows the actuators to transmit assistive torques to knee joint when the knee is flexed and transmit assistive torques to ankle joint when the knee is fully extended. Design concepts and component descriptions were elaborated for this underactuated soft exoskeleton, including kinematics, actuation, etc. The performance of the underactuated soft exoskeleton was evaluated by a series of experiments. With effective assistance from the proposed exoskeleton, the power of the knee was reduced by 6.40%, compared to without exoskeleton condition. Meanwhile, the moment and power of the ankle also demonstrated significant reduction when comparing the exoskeleton powered condition to the without exoskeleton condition (reduced by 11.61% and 45.44%, respectively). This underactuated soft exoskeleton represents a feasible solution to apply sequential assistance to the knee and ankle, and the experimental results validated its effectiveness in assisting the motions which have chronological moments at each lower-limb joint.