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Vertebrates, including amphibians, reptiles, birds, and mammals, with their ability to change the stiffness of the spine to increase load-bearing capability or flexibility, have inspired roboticists to develop artificial variable-stiffness spines. However, unlike their natural counterparts, current robotic spine systems do not display robustness or cannot adjust their stiffness according to their task. In this paper, we describe a novel variable-stiffness tensegrity spine, which uses an active mechanism to add or remove a ball-joint constrain among the vertebrae, allowing transition among different stiffness modes: soft mode, global stiff mode, and directional stiff mode. We validate the variable-stiffness properties of the tensegrity spine in experimental bending tests and compare results to a model. Finally, we demonstrate the tensegrity spine system as a continuous variable-stiffness manipulator and highlight its advantages over current systems.