Cellular material (emulsions, foams) made out of two different phases, one dispersed in another, may coarsen with time through coalescence, which is the rupture of the thin liquid film that separates two adjacent droplets or bubbles. In this Letter, we study destruction through coalescence of a model cellular material: a monodisperse soap foam. We report the existence of a sharp destabilization threshold controlled only by the liquid fraction of the foam at which the rate of coalescence increases dramatically. We point out a coupling between drainage and coalescence. We suggest that the rearrangements of the bubbles during the drainage of the foam induce an increase of the area of the bubbles which decreases temporarily the amount of adsorbed surfactant by unit area and weakens the interfaces.