Lately, extensive research efforts have been dedicated to the design of clustering algorithms to organize nodes in Vehicular Ad Hoc Networks (VANETs) into sets of clusters. However, due to the dynamic nature of VANETS, nodes frequently joining or leaving clusters jeopardize the stability of the network. The impact of these perturbations becomes worse on network performance if these nodes are cluster heads. Therefore, cluster stability is the key to maintain a predictable performance and has to consider reducing the clustering overhead, the routing overhead and the data losses. In this paper, we propose a new stability-based clustering algorithm (SBCA), specifically designed for VANETs, which takes mobility, number of neighbors, and leadership (i.e., cluster head) duration into consideration in order to provide a more stable architecture. Extensive simulations show that the proposed scheme can significantly improve the stability of the network by extending the cluster head lifetime longer than other previous popular clustering algorithms do.