The process of capillary breakup of a thread of Newtonian liquid is considered theoretically in the simplest case where the thread is surrounded by an inviscid, dynamically passive gas. The goal is to remove the singularities inherent in the known solutions to the problem obtained in the framework of the standard model and explain some puzzling qualitative features of the process observed in experiments. The analysis is based on the idea that, since the known solutions indicate that the rate at which fresh free-surface area is created tends to infinity as breakup is approached, one has that the surface tension, whose relaxation to equilibrium is always associated with a small but finite relaxation time, is bound to deviate from its equilibrium value in the process of breakup. This gives rise to a surface-tension gradient which starts to pull the liquid thread apart (the flow-induced Marangoni effect), whilst the role of the capillary pressure-driven squeezing of the liquid out of the neck diminishes as the surface tension in the minimal cross-section decreases. An earlier developed theory incorporating the interface formation process is applied without any ad hoc alterations and analysed in the framework of the slender-jet approximation. The resulting solution is singularity-free and allows one to describe some previously unexplained features of experiment by Kowalewski (1996, Fluid Dyn. Res., 17, 121).