Numerous biological studies of body and/or caudal fin fish have powerfully identified that the fish body's passive properties, such as stiffness and its distribution, contribute to the fish's outstanding swimming performance. In this study, we develop a free-swimming tensegrity fish robot (TenFiBot) to help scholars experimentally improve their understanding of the role of stiffness distribution in fish-like swimming. We detail the overall structure and parameters of TenFiBot, which can achieve body stiffness distribution with high stiffness controllability over a wide range. The effective variable stiffness principle is based on antagonistic stiffness that results from the prestress of tension structures in a kinematically singular configuration. We validate the effectiveness of variable stiffness for one variable-stiffness tensegrity joint (VSTJ) and demonstrate that the virtual rotational centers are always in the acceptable range during stiffness variation. We exemplify some cases of the stiffness distribution to verify the validity of body stiffness variation. The experimental results show that swimming performance, such as swimming velocity, varied dramatically when the stiffness distribution was varied for the robotic fish. The swimming performance of the robotic fish can be enhanced tremendously by changing the stiffness distribution. Moreover, the VSTJs in the fish body have great potential in constructing compliant robotic arms and snake-like robots.