We study a new relaxation for a two-dimensional optimal design problem in conductivity consisting of determining how to mix two given conducting materials in order to minimize the amount of one of them, subject to a constraint on the efficiency of the conducting properties of the mixture. Our approach here is different from that obtained in [R. V. Kohn and G. Strang, Comm. Pure Appl. Math., 39 ( 1986), pp. 113 - 137, 139 - 182, 353 - 377], and is based on a local reformulation of the optimal design problem by means of the introduction of new potentials. The concept of constrained quasiconvexification is used in an important way.