We consider a fluid crossing a zone of rapid density change, so thin that it can be considered as a density jump interface. In this case, the normal velocity undergoes a jump. For a Newtonian viscous fluid with low Reynolds number (creeping flow) that keeps its rheological properties within the interface, we show that this implies that the traction cannot be continuous across the density jump because the tangential stress is singular. The appropriate jump conditions are established by using the calculus of distributions, taking into account the curvature of the interface as well as the density and viscosity changes. Independently of any intrinsic surface tension, a dynamic surface tension appears and turns out to be proportional to the mass transfer across the interface and to a coefficient related to the variations of density and viscosity within the interface. Explicit solutions are exhibited to illustrate the importance of these new jump conditions. The example of the Earth's inner core crystallisation is questioned.