The sparse signal recovery in standard compressed sensing (CS) requires that the sensing matrix is exactly known. The CS problem subject to perturbation in the sensing matrix is often encountered in practice and has attracted interest of researches. Unlike existing robust signal recoveries with the recovery error growing linearly with the perturbation level, this paper analyzes the CS problem subject to a structured perturbation to provide conditions for stable signal recovery under measurement noise. Under mild conditions on the perturbed sensing matrix, similar to that for the standard CS, it is shown that a sparse signal can be stably recovered by ℓ1 minimization. A remarkable result is that the recovery is exact and independent of the perturbation if there is no measurement noise and the signal is sufficiently sparse. In the presence of noise, largest entries (in magnitude) of a compressible signal can be stably recovered. The result is demonstrated by a simulation example.