We report the first observation of the disappearance of flow in heavy-ion collisions. This is accomplished by measuring the excitation function of the average in-plane transverse momentum for the symmetric system {sup 139}La+{sup 139}La, using beam energies of 130, 70, and 50 MeV/nucleon. The observation is indicative of a change from dominantly repulsive to attractive scattering. We also present the results of calculations performed with the Boltzmann-Uehling-Uhlenbeck equation which support the concept of vanishing flow for this system in the energy region between 30 and 50 MeV/nucleon.