After a droplet has broken away from a slender thread or jet of liquid, the tip of the thread or jet recoils rapidly. At the moment of break-off, the tip of the thread/jet is observed to have the shape of a cone close to the bifurcation point. In this paper, we study the evolution of an ideal fluid which is initially conical, where the only force acting on the fluid is due to surface tension. We find an asymptotic solution to the problem in terms of the aspect ratio of the cone which is assumed to be small. Using a similarity transformation, which is valid for small times after the bifurcation, we identify a rapidly oscillating non-linear wave which propagates away from the tip, as observed in experiments.