In this contribution, a mathematical model is presented for the flow of molten salt in a circular pipe. The fluid density, thermal conductivity, and viscosity are all assumed to be temperature dependent. We show how to derive a new closed-form approximate solution to this problem, valid for low Reynolds numbers and high axial temperature gradients. This regime is chosen to challenge the ability of the model to obtain the correct solution for significant changes in fluid properties. The accuracy of this simplified solution is tested against a numerical solution of the full set of governing equations, and it is found that the maximum error is less than 1.5% for the temperature field and 6% for the velocity field, for Reynolds numbers less than 100. This closed-form solution allows us to gain insight into the effects of various physical parameters upon the flow properties and thermal behavior of the molten salt, and this is illustrated in the results. The solution presented here also serves as a benchmark for numerical solutions of the full model.