Community detection and link prediction are highly dependent since knowing cluster structure as a priori will help identify missing links, and in return, clustering on networks with supplemented missing links will improve community detection performance. In this paper, we propose a Cluster-driven Low-rank Matrix Completion (CLMC), for performing community detection and link prediction simultaneously in a unified framework. To this end, CLMC decomposes the adjacent matrix of a target network as three additive matrices: clustering matrix, noise matrix and supplement matrix. The community-structure and low-rank constraints are imposed on the clustering matrix, such that the noisy edges between communities are removed and the resulting matrix is an ideal block-diagonal matrix. Missing edges are further learned via low-rank matrix completion. Extensive experiments show that CLMC achieves state-of-the-art performance.