We study the optimal design of a conductance network as a means for synchronizing a given set of oscillators. By considering the conductance that connects two oscillators as the measure of the amount of communication between them, we formulate an optimal control problem that addresses the trade-off between synchronization performance and communication. Additionally, we promote the sparsity of the network by penalizing the number of interconnection links. For identical oscillators, we establish convexity and show that the design problem can be formulated as a semidefinite program. For non-identical oscillators, that can be considered as perturbations around a central (average) oscillator, we show that it is meaningful to design an optimal conductance network by assuming that all oscillators are identical to the central oscillator. Finally, for special classes of oscillator networks we derive explicit formulas for the optimal conductance values.