The ability to extend the operating life of a pressurized water reactor depends in part on the ability of the reactor pressure vessel to withstand thermal shock concurrent with significant pressure. If during the course of a small-break loss-of-coolant accident (SBLOCA), the primary-side pressure is reduced sufficiently, cold make-up water is supplied to the cold leg by the emergency core cooling system. If incomplete mixing occurs between the cold injected water and the hot water in the primary circuit, a stream of cool water flows along the bottom of the cold leg into the downcomer. There, the cool water forms a downward-flowing buoyant plume surrounded by the hot water in the downcomer. The time-dependent spatial distributions of the temperatures and heat transfer coefficients on the inside surface of the reactor pressure vessel are important in determining compliance with regulatory requirements. The simulation of the mixing in the cold leg and downcomer is typically performed with flow-mixing computer codes, most of which use either computational fluid dynamics techniques or mechanistic models. The computer code used for this work, called KWU-MIX, makes use of mechanistic models. In previous works, the uncertainties in parameters associated with the most important phenomena that contribute to the temperature distributions were quantified by comparing experimentally derived values of the parameters with values from the mechanistic models. In this work, those uncertainties are propagated through the flow-mixing code in order to quantify the uncertainty in the calculated temperature distributions. An example of the propagation of uncertainties is given for conditions typical of a SBLOCA. Random values from each of the uncertainty distributions for the parameters of all of the most important phenomena were selected for each of 100 simulations of the typical accident conditions. The results of the 100 simulations were analyzed statistically in order to quantify the best-estimate temperature distribution and its uncertainty. The resulting best-estimate temperature distribution and its uncertainty were compared with experimental data obtained in the Upper Plenum Test Facility at the same typical accident conditions. The results of the comparison show that the uncertainty in the calculated temperature distribution bounds the experimental values.