We study a Rayleigh-like instability of an oil in water cylindrical "droplet" in the presence of excess surfactant. The surfactant monolayer is modeled by an interfacial bending energy with a nonvanishing spontaneous curvature C0. We find that, even when surfactant reduces the interfacial tension to a vanishingly small value, the bending energy with spontaneous curvature can destabilize the cylinder if its radius R is greater than C0-1. The cylinder deforms into "beads" which can eventually break up into spherical droplets whose radius we predict. Unlike in the classical Rayleigh case, we find that the total interfacial area is usually increasing. Using our theory, we discuss the kinetics of the transition from a microemulsion phase of swollen cylindrical micelles to a droplet phase, following a sudden increase in the spontaneous curvature parameter.