A phase of low brightness during the radiation emission from the positive plasma column is discovered in the initial stage of the evolution of a low-pressure glow discharge in helium and in the mixture He-N₂. The duration of this "dark" phase ranges from tens of microseconds to several milliseconds; the radiation intensity during the dark phase is less than that in the steady state by one to three orders of magnitude. At the same time, the value of the discharge current is close to the steady one. A similar phase was observed in the time dependence of the electric field strength. A model is proposed for explaining the observed effect of an excess of metastable atoms He* appearing in the beginning of the pulse that produces the Penning electrons. The measured time dependences of E/N and of the concentration of He* are in good agreement with the proposed model.