The paper deals with the stiffness analysis and stability study of equilibrium configurations for dual-triangle tensegrity mechanism, which is actuated by adjusting elastic connections between the triangle edges. For this mechanism, the torquedeflection relation was obtained as a function of control inputs and geometric parameters. It was proved that the mechanism can has either a single or three equilibrium configurations that can be both stable and unstable. Corresponding conditions of stability were found allowing user to choose control inputs ensuring the mechanism controllability. The obtained results are confirmed by the simulation examples presented in the paper.