The principle of minimum entropy production says that the steady state of an irreversible process, i.e., the state in which the thermodynamic variables are independent of the time, is characterized by a minimum value of the rate of entropy production. This theorem, due to Prigogine, is proved by the methods of statistical mechanics for a particular process---the flow of matter and energy through a narrow tube connecting two containers of an ideal gas. The two containers are maintained at slightly different temperatures. The resultant form for the entropy production in the steady state, and the method used in the proof, give additional insight into the significance of the principle of minimum entropy production.