Natural circulation is important for the long-term cooling of light water reactors in off-normal conditions, and it is therefore important to understand the numerical behavior of reactor safety codes used to simulate flows under those conditions. While the methods and models in these codes have been studied in some detail, the impact of the weight force term on the numerical behavior has been largely ignored. The dynamic and numerical stability of the one-dimensional, single-phase-flow equations are examined for natural-circulation problems. It is shown that the presence of the weight force in the momentum equation results in a minimum value of the frictional loss coefficient for the equations to be stable. It is further shown that the numerical solution is unstable unless this dynamic stability limit is satisfied. The stability limits developed are verified by numerical solution of the single-phase-flow equations under natural-circulation conditions.