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We present a fast robust recursive least-squares (FRRLS) algorithm based on a recently introduced new framework for designing robust adaptive filters. The algorithm is the result of minimizing a cost function subject to a time-dependent constraint on the norm of the filter update. Although the characteristics of the exact solution to this problem are known, there is no closed-form solution in general. However, the approximate solution we propose is very close to the optimal one. We also present some theoretical results regarding the asymptotic behavior of the algorithm. The FRRLS is then tested in different environments for system identification and acoustic echo cancellation applications.