Presently the positron ring dedicated to positronium generation and called the Low Energy Positron Toroidal Accumulator (LEPTA) is under construction at JINR (Dubna). The main application of this ring is the generation of a monoenergetic and directed flux of positronium atoms and its use in experiments. Positronium is generated in interactions of positrons, circulating inside a storage ring accumulator, with free electrons of the cooling electron beam that have velocities very close to the positron velocities. This allows us to obtain a high positronium flux with small angular and velocity spreads of the atoms and provides a significant advantage for the proposed arrangement of experiments, so-called positronium-in-flight setups, as compared with traditional approaches in which positronium was generated in targets. In particular, the precision in measuring positronium parameters (lifetime, the probability of decays with momentum conservation, charge invariant violation (CPT violation), fine structure of the positronium spectrum, and Lamb-shift measurements) can be enhanced by several orders of magnitude. Moreover, some experiments that are unrealistic within traditional schemes become feasible with the proposed facility. PACS No.: 36.10Dr