We consider numerical approximations for solving a new, Allen–Cahn type ternary phase-field model. We first formulate a phase-field model from an energetic variational formulation based on the L2-gradient flow and add three nonlocal Lagrange multipliers to the system to conserve the volume for each phase. Then, by combining the SAV approach with the stabilization technique, where a crucial linear stabilization term is added to enhance the stability and keep the required accuracy thus allowing large time steps, we arrive at a decoupled, linear, non-iterative, and volume-conserved scheme. This scheme is unconditionally energy stable and requires solving only four linear second-order elliptic equations with constant coefficients. We further prove energy stability and present numerous 2D and 3D numerical simulations to show accuracy and stability.