Birds are so stable that they can rest and even sleep standing up. We propose that stable static balance is achieved by tensegrity. The rigid bones can be held together by tension in the tendons, allowing the system to stabilize under the action of gravity. We used the proportions of the bird's osteomuscular system to create a mathematical model. First, the extensor muscles and tendons of the leg are replaced by a single cable that follows the leg and is guided by joint pulleys. Analysis of the model shows that it can achieve balance. However, it does not match the biomechanical characteristics of the bird's body and is not stable. We then replaced the single cable with four cables, roughly corresponding to the extensor groups, and added a ligament loop at the knee. The model is then able to reach a stable equilibrium and the biomechanical characteristics are satisfied. Some of the anatomical features used in our model correspond to innovations unique to the avian lineage. We propose that tensegrity, which allows light and stable mechanical systems, is fundamental to the evolution of the avian body plan. It can also be used as an alternative model for bipedal robots.