This work presents a mathematical analysis to solve the Riemann problem for a two-phase gas–liquid mixture. For this purpose, we use the four-equation homogeneous equilibrium model with the Stiffened equation states. The exact solution of the Riemann problem is obtained by providing an explicit formulation of the different elementary waves: rarefaction, shock, and contact discontinuity. The validation of the proposed mathematical approach is confirmed by the good agreement found between the analytical solutions and the calculated results obtained by the numerical schemes of Mac Cormack and Godunov for various problems such as the Riemann problem, the partial Riemann problem for the treatment of boundary conditions, and the water hammer problem caused by a rapid closing valve.