A balloon-borne superconducting solenoidal magnet spectrometer, BESS, has been designed and developed to investigate cosmic-rays and particle-astrophysics. Search for primordial antimatter in cosmic-rays and observation of gamma-rays are primary objectives to study early history of the Universe. The cylindrical spectrometer configuration with a thin superconducting solenoid magnet is optimized to provide a large geometrical acceptance of 0.5-1.0 m2sr for these scientific objectives. The spectrometer system has been completed and is planned to be launched for the first scientific flight from Lynn Lake, Canada, in 1993. This report describes the progress of the spectrometer development and discuss about its capability to search for primordial antimatter in cosmic rays.