One-dimensional particle-in-cell and Monte-Carlo (FLUKA) simulation methods were used together, in order to investigate the production of positrons in lead targets, illuminated by a short-pulse high-intensity laser. The study is focused on lead targets of 1 mm thickness and more and pulses of intensity in the range 1019–1022 W/cm2. The calculations provided an estimate of an absolute number of positrons and the ratio of electrons and positrons emerging from the target. The thickness of the target is scaled in order to find an optimal thickness that could provide a neutral electron-positron beam.