It is generally accepted that the equation of state of the Debye and Hückel theory, originally developed for strong electrolytes basing on the classical statistical mechanics, is applicable to the high temperature plasmas in thermal equilibrium. However, if we were to apply the classical statistical mechanics to the fully ionized plasmas, the partition function would diverge because of the short range attraction between a pair of positive and negative charges, and there is a doubt whether the contribution of this infinite attraction may not overcome the contribution of the Debey-Hückel term. In this paper the equation of state of high temperature plasmas is investigated in consideration of the quantum mechanics; and it is shown that the Debey-Hückel approximation surely applies to the plasmas of low density of the order 1015 ∼ 1017 or so at high temperatures where λe ≲a0/Z, where Z is the charge of a nucleus, λe the de Broglie wave length of an electron and a0 the Bohr radius. This result is obtained by reducing the problem to that of a suitable classical gas and confirming that the contribution of the watermelon terms–which is considered as the leading correction to the Debye-Hückel approximation–is negligible compared with that of the ring terms considered in the Debye-Hückel approximation.