This paper reports the solution to the fundamental problem of how to maximize the mechanical power extracted from a hot single-phase stream when the total heat transfer area bathed by the stream is constrained. It is shown that the optimization has two degrees of freedom : the shape of the stream temperature distribution as a function of the length (x) traveled along the heat transfer surface, and the position of this distribution on the absolute temperature scale. The optimal stream temperature distribution is exponential in x, and so is the temperature distribution along the hot end of the system that converts the heat transfer into mechanical power. At any x, the temperature difference across the heat exchanger is proportional to the local absolute temperature. Similar conclusions are reached for the cold end heat exchanger, when the power system rejects heat to a cold single-phase stream. It is shown that the optimal solution can be implemented in practice by using two counterflow heat exchangers. Each counterflow is imbalanced to a degree recommended by thermodynamic optimization. The effect of the sizes and capacity rates of the two heat exchangers is documented.