A one-dimensional inviscid solution for flow through a compliant tube with a stenosis is presented. The model is used to represent an artery with an atherosclerotic plaque and to investigate a range of conditions for which arterial collapse may occur. The coupled equations for flow through collapsible tubes are solved using a Runge-Kutta finite difference scheme. Quantitative results are given for specific physiological parameters including inlet and outlet pressure, flow rate, stenosis size, length and stiffness. The results suggest that high-grade stenotic arteries may exhibit collapse with typical physiological pressures. Critical stenoses may cause choking of flow at the throat followed by a transition to supercritical flow with tube collapse downstream. Greater amounts of stenosis produced a linear reduction of flow rate and a shortening of the collapsed region. Changes in stenosis length created proportional changes in the length of collapse. Increasing the stiffness of the stenosis to a value greater than the nominal tube stiffness caused a greater amount of flow limitation and more negative pressures, compared to a stenosis with constant stiffness. These findings assist in understanding the clinical consequences of flow through atherosclerotic arteries.