A situation in which a dilffusion flame reaches an end at some position in a medium of non-premixed reactants is studied. The mixing of reactants that takes place ahead of the diffusion flame leads to the formation of a “triple-flame”, a structure which consists of a fuel-rich premixed flame, a fuel-lean premixed flame, and a diffusion flame that starts where the two premixed flames meet. An important property of such an end-point is its ability to propagate. The limits of low heat release, unit Lewis number and large Zeldovich number are considered. The structure of the triple-flame and the unique relationship between propagation speed and transverse mixture Traction gradient are computed numerically. For the range of values considered here, the end of the diffusion flame is shown to extend itself at a rate that can be substantially reduced, but that remains positive as the gradient of the mixture fraction is increased.