The free energy of a ternary system with a self-organization property includes an interface energy and a longer ranging, inhibitory interaction energy. In a planar domain, if the two energies are properly balanced and two of the three constituents make up an equal but small fraction, the free energy admits a local minimizer that is shaped like a perturbed double bubble. Most difficulties in the proof of this result are related to the triple junction phenomenon that the three constituents of the ternary system meet at a point. Two techniques are developed to deal with the triple junction. First, one defines restricted classes of perturbed double bubbles. Each perturbed double bubble in a restricted class is obtained from a standard double bubble by a special perturbation. The two triple junction points of the standard double bubble can only move along the line connecting them, in opposite directions, and by the same distance. The second technique is the use of the so called internal variables. These variables derive from the more geometric quantities that describe perturbed double bubbles in restricted classes. The advantage of the internal variables is that they are only subject to linear constraints, and perturbed double bubbles in a restricted class represented by internal variables are elements of a Hilbert space. A local minimizer of the free energy in each restricted class is found as a fixed point of a nonlinear equation by a contraction mapping argument. The second variation at the fixed point within its restricted class is positive. This perturbed double bubble satisfies three of the four equations for critical points of the free energy. The unsolved equation is the 120 degree angle condition at triple junction points. Then perform another minimization among the local minimizers from all restricted classes. A minimum of minimizers emerges and solves all the equations for critical points.