The Benjamin-Feir instability is a modulational instability in which a uniform train of oscillatory waves of finite amplitude loses energy to a small perturbation of waves with nearly the same frequency and direction. The concept is well established in water waves, in plasmas, and in optics. Even so, we show that any amount of dissipation (of a certain type), no matter how small, stabilizes this instability. We confirm our analytical predictions with laboratory experiments on waves in deep water. Finally, we reexamine previous experimental studies of an instability that we claim does not exist in the presence of dissipation.