We analyze the Basis Pursuit recovery of signals when observing K-sparse data with general perturbations (i.e., additive, as well as multiplicative noise). This completely perturbed model extends the previous work of Cand`es, Romberg and Tao on stable signal recovery from incomplete and inaccurate measurements. Our results show that, under suitable conditions, the stability of the recovered signal is limited by the noise level in the observation. Moreover, this accuracy is within a constant multiple of the best-case reconstruction using the technique of least squares.