This paper discusses the existence of ion-acoustic solitons in a plasma consisting of hot and cool ions, and hot electrons, where the words “hot” and “cool” reflect the ratio of thermal effects to inertial effects. For the hot species general polytropic pressure relations are considered, as well as the isothermal limit, γ=1 (Boltzmann distribution). It is shown that negative potential solitons are not supported. Existence domains in parameter space (effective soliton Mach number and fractional cool ion density) for positive potential solitons are found, the limits being imposed by physical effects such as vanishing densities. Numerical results are given for a typical value of γ>1 and for γ=1, for three representative values of the hot ion to electron temperature ratio.