In massive network data analysis, especially online social network analysis, the complete network dataset is often difficult to obtain due to the huge cost in collecting and storing data. In order to recover missing data from sampled networks, we consider the network completion problem, which has attracted much attention from both academia and industry. In this letter, the network completion problem of sparse networks is solved by a proposed discrete-constrained nuclear-norm minimization (DNM) method. It is based on the sparsity of the number of nonzero elements and singular values, which leads to an optimization problem. Since the problem is NP-hard, relaxation and adjustment are applied to make it convex. The DNM method can be applied in many practical networks, as many real-world complex networks are sparse. The simulation results on real-world online social networks and artificially generated random networks indicate that the proposed DNM method outperforms many existing network completion methods.