We consider a symmetric $alpha$-stable model for heavy tailed time series, which allows for some dependence structure (or memory) in the data. In this context, an estimator for the tail index is $alpha$ is presented, which has a rapid rate of convergence -in particular $Op(n^{-1/2})$ -which is robust under intermediate memory. There is no need for blocking or tuning parameters for this estimator. Small sample results and full asymptotics are provided in this paper, and simulation studies on various $alpha$-stable data sets are given as well.