We present the results of our study of a weakly ionized decaying helium plasma with a small admixture of neon by means of kinetic spectroscopy. The experimental conditions are a helium pressure of 10–38 Torr, an electron density [e] ≤ 2 × 10¹¹ cm^–3, and a neon density [Ne] ≤ 2 × 10^–4 [He]. The recombination processes of [Ne]≤ 2 × 10^-4[He], HeNe⁺, and Ne⁺ ions with electrons have been identified by the decaying-plasma radiation. The neon afterglow spectrum at [e] ≤ 10¹¹ cm^–3 is shown to be formed mainly by dissociative recombination of Ne⁺₂ and HeNe⁺ ions in the ground vibrational state v = 0. The early afterglow of a number of atomic neon lines is formed by the well-known process of excitation transfer from helium atoms in metastable states He(2³S₁, 2¹S₀). At [e] > 10¹¹ cm^–3, the afterglow spectrum is enriched due to the collisional-radiation recombination of Ne⁺ ions. In the recombination stage of the afterglow at room electron temperature, the existence of a clear upper bound on the excitation energy of the atomic neon levels being populated is characteristic of the dissociative recombination of both molecular ions. For Ne⁺₂ this is the 3p₁ level (in Paschen notation) with an energy of 20.369 eV, with the levels closest to Ne⁺₂(v=0) in energy being 3p₃ and 3p₂ with energies of 20.26 and 20.3 eV, respectively; for HeNe⁺, this is the 5d^'₁ level with an energy of about 21.02 eV. These values are proposed to be taken as the energies of Ne⁺₂ and HeNe⁺ ions in the ground vibrational state.