A decaying weakly ionized helium plasma [n_e=(0.2-1.1)·10¹¹ cm^-3, p=(40-70) mm Hg] was studied experimentally. It is shown that the experimental time dependences of the intensities of atomic lines and molecular bands in the afterglow phase can be explained if the vibrational kinetics of He₂⁺ ions is included in the analysis. Analysis of the measurements shows that for n_e ≲ 10¹¹ cm^-3 and N_a ≳ 10¹⁸ cm^-3 deexcitation of He₂⁺ ions occurs primarily as a result of inelastic collisions with helium atoms. Based on the experimental data, an approximate value was obtained for the rate constants of the vibrational relaxation of molecular helium ions ≈10^-16 cm³/sec. These results are used for making a qualitative analysis of the distribution of He₂⁺ ions over the vibrational states in the discharge phase.