Oil and water can only be mixed by dispersing droplets of one fluid in the other. When two droplets approach one another, the thin film that separates them invariably becomes unstable, causing the droplets to coalesce. The only known way to avoid this instability is through addition of a third component, typically a surfactant, which stabilizes the thin film at its equilibrium thickness. We report the observation that a thin fluid film of oil separating two water droplets can lead to an adhesive interaction between the droplets. Moreover, this interaction prevents their coalescence over timescales of several weeks, without the use of any surfactant or solvent.