Abstract We report the first laboratory experiment dealing with the interaction of a cosmic dust simulant with positrons emitted from a ²²Na radioisotope. Measurements of a charge of micrometer SiO₂ dust grains with an accuracy of one elementary charge e revealed +1 e steps due to positron annihilation inside the grain. The observed average rate of these charging events agrees well with prediction of a model based on the continuous slowing down approximation of energetic of positrons inside the grain. Less frequent charge steps larger than +1 e were attributed to emission of secondary electrons during positron slowing down. The determined coefficient of secondary electron emission is approximately inversely proportional to the grain radius. The experimental results led us to the formulation of a possible scenario of interstellar dark clouds charging.