  In this paper, we prove a general result concerning finite-range, attractive
interacting particle systems on $\{-1, 1\}^{\mathbb{Z}^d}$. If the particle
system has a unique stationary measure and, in a precise sense, relaxes to this
stationary measure at an exponential rate then any small perturbation of the
dynamics also has a unique stationary measure to which it relaxes at an
exponential rate. To augment this result, we study the particular case of
Glauber dynamics for the Ising model. We show that for any non-zero external
field the dynamics converges to its unique invariant measure at an exponential
rate. Previously, this was only known for $\beta<\beta_c$ and $\beta$
sufficiently large. As a consequence, Glauber dynamics is stable to small,
non-equilibrium perturbations in the entire uniqueness phase.
