Consider the problem of recovering a vector from corrupted measurements of linear combinations of its components. It is known that when only a relatively small fraction of the measurements is corrupted, and the rest is error-free, the vector can be exactly recovered by l1-norm minimization. We introduce a new, robust, error correction mechanism that covers the case when a fraction of the measurements is corrupted by arbitrary, eventually adversarial, errors and additionally all the measurements carry some noise. We show that, by solving a nonsmooth convex minimization problem, it is possible to recover the least-squares estimate of the vector as if it was contaminated with noise only. Moreover, we show that the fraction of arbitrary errors that the estimator can manage is exactly the same as that the l1-norm minimization can face in the noiseless case. Finally, we present a globally convergent forward-backward algorithm for computing our estimator.