Suppose we want to recover a vector x0isinRmfrom incomplete and contaminated measurements y=Ax0+e, where A is a p by m matrix, p2=epsiv. In previous work, Donoho et al. (1998), Candes et al. (Commun. Pure Appl.) have shown that if the signal x0is sufficiently sparse, it can be recovered stably by lscr1-minimization. More precisely, the vector x#with minimal lscr1norm, constrained by parAx#-ypar2lesepsiv, satisfies parx#-x0par2lesCepsiv. In the present correspondence it is shown that one can also choose a regularization approach, in which one minimizes the functional parAx-ypar22+muparxpar1, without additional constraints; its minimizer x* is likewise a good estimate for x0, with a bound on parx*-x0par2proportional to epsiv.