The problem of a polymer molecule whose two ends reside on opposite sides of a membrane or partition separating two solutions is solved exactly in the limit of no self-excluded volume. The monomers can go from one side of the membrane to the other only by threading serially through one hole in the membrane. The ends can be free, confined to run freely on the membrane surfaces, or be fixed to specific points on the membrane. It is found that the equilibrium thermodynamic phase transition is first order in all cases so that slight changes in pH, ionic strength, or temperature can move the polymer from being completely on one side of the membrane to being completely on the other side. Application to two biological problems are suggested: (1) the breaching of cell walls by the nuclear material of T2 bacteriophages, and (2) the transport of drugs that are affixed to these translocating polymers. The relation of this newly discovered transition to four other phase transitions that occur in isolated macromolecules (helix-random coil; equilibrium polymerization; polymer collapse; surface adsorption) is briefly discussed. (C) 1997 American Institute of Physics.