A jumping leg with one degree of freedom (DOF) is characterized by high rigidity and simple control. However, robots are prone to motion failure because they might tip over during the jumping process due to reduced mechanism flexibility. Mechanism design, configuration optimization, and experimentation were conducted in this study to achieve jumping stability for a bioinspired robot. With locusts as the imitated object, a one-DOF jumping leg mechanism was designed taking Stephenson-type six-bar mechanism as reference, and kinematic and dynamic models were established. The rotation angle of the trunk and the total inertia moment were used as stability criteria, and the sensitivity of different links to the target was analyzed in detail. With high-sensitivity link lengths as the optimization parameters, a configuration optimization method based on the particle swarm optimization algorithm was proposed in consideration of the different constraint conditions of the jumping leg mechanism. Optimization results show that this method can considerably improve optimization efficiency. A prototype of the robot was developed, and the experiment showed that the optimized trunk rotation angle and total inertia moment were within a small range and can thus meet the requirements of jumping stability. This work provides a reference for the design of jumping and legged robots.