The possibility of a high intensity low energy positron beam production is discussed. The proposed Positron String Trap (PST) is based on the principles and technology of the Electron String Ion Source (ESIS) developed in JINR during the last decade. A linear version of ESIS has been used successfully for the production of intense highly charged ion beams of various elements. Now the Tubular Electron String Ion Source (TESIS) concept is under study and this opens really new promising possibilities in physics and technology. In this report, we discuss the application of the tubular-type trap for the storage of positrons cooled to the cryogenic temperatures of 0.05 meV. It is intended that the positron flux at the energy of 1–5 eV, produced by the external source, is injected into the Tubular Positron Trap which has a similar construction as the TESIS. Then the low energy positrons are captured in the PST Penning trap and are cooled down because of their synchrotron radiation in the strong (5–10 T) applied magnetic field. It is expected that the proposed PST should permit storing and cooling to cryogenic temperature of up to 5×109 positrons. The accumulated cooled positrons can be used further for various physics applications, for example, antihydrogen production.