In this paper an analysis is given for the conditions under which compressive and rarefactive ion-acoustic solitons may exist in a plasma containing two electron populations, one hot and the other cool. Compressive solitons exist in the collective Mach number range $1\,{<}\,M\,{<}\,M_{\rm c}$, which, for a given abundance ratio of the cool population, translates into an annular region in the ion-acoustic Mach numbers space. The upper limit $M_{\rm c}$ corresponds to a degenerate sonic point for the protons in which they tend to an infinite compression and are brought to rest at the wave centre. To form a charge neutral point in the rarefactive regime (a prerequisite to the existence of rarefactive solitons) it is necessary that the ‘scale height’ of the hot electrons be sufficiently large that their number density, which is nearly constant, approximately balances that of the protons, before the cool electrons experience their ‘atmospheric lid’. Furthermore, a sufficient condition for the existence of rarefactive solitons is found by requiring a solution to the wave structure equation at the ‘lid’. The case of very hot electrons (highly sub-ion-acoustic) is considered, and various parameter regimes in which compressive and rarefactive solitons exist are evaluated in this super-hot electron approximation, both for the polytropic case, $\gamma \neq 1$, and the isothermal case, $\gamma = 1$. This approximation is formally equivalent to the problem of a bi-ion plasma, including a super-massive negative species.