This paper examines simultaneous design of the network topology and the corresponding edge weights in the presence of a cardinality constraint on the edge set. Network properties of interest for this design problem lead to optimization formulations with convex objectives, convex constraint sets, and cardinality constraints. This class of problems is referred to as the cardinality-constrained optimization problem (CCOP); the cardinality constraint generally makes CCOPs NP-hard. In this paper, a customized alternating direction method of multipliers (ADMM) algorithm aiming to improve the scalability of the solution strategy for large-scale CCOPs is proposed. This algorithm utilizes the special structure of the problem formulation to obtain closed-form solutions during each iterative step of the corresponding ADMM updates. We also provide a convergence proof of the proposed customized ADMM to a stationary point under certain conditions. Simulation results illustrate that the customized ADMM algorithm has a significant computational advantage over existing methods, particularly for large-scale network design problems.