The annihilation process for a superheavy monopole-antimonopole pair connected by a relatively light magnetic flux tube (which is possibly relevant to the cosmological monopole problem) is studied. Three energy-loss mechanisms are considered in some detail. The first two are the radiation of photons and weak bosons due to the linear and Coulomb parts of the potential, respectively. Then the energy loss due to the frictional drag of monopoles moving through a relativistic plasma is analyzed. We argue that for the parameters relevant to standard theories this last mechanism dominates by many orders of magnitude and leads to an almost instantaneous annihilation at the time confinement takes place. The implications of this work for the dilution of the baryon to entropy ratio are described.Permanent address: Department of Physics, University of Pennsylvania, Philadelphia, PA 19104, USA.