Convex clustering has drawn recent attentions since it nicely addresses the instability issue of traditional non-convex clustering methods. Although its computational and statistical properties have been recently studied, the performance of convex clustering has not yet been investigated in the high-dimensional clustering scenario, where the data contains a large number of features and many of them carry no information about the clustering structure. In this paper, we demonstrate that the performance of convex clustering could be distorted when the uninformative features are included in the clustering. To overcome it, we introduce a new clustering method, referred to as Sparse Convex Clustering, to simultaneously cluster observations and conduct feature selection. The key idea is to formulate convex clustering in a form of regularization, with an adaptive group-lasso penalty term on cluster centers. In order to optimally balance the trade-off between the cluster fitting and sparsity, a tuning criterion based on clustering stability is developed. Theoretically, we obtain a finite sample error bound for our estimator and further establish its variable selection consistency.