The new AEGIS (Antimatter Experiment: Gravity, Interferometry, Spectroscopy) Experiment will be installed in the Antiproton Decelerator hall at CERN. The main goal is to measure the Earth's gravitational acceleration of antihydrogen atoms. The experiment consists of two high-homogeneity solenoids placed on the same axis. The 5 T magnet is part of a cylindrical Penning trap to catch and to accumulate antiprotons delivered by the decelerator. The antihydrogen is then produced in the 1 T region where sub-kelvin antiproton temperatures provided by the dilution refrigerator are required to form a slowly-moving beam of antihydrogen. The helium bath cooled superconducting magnets; the different traps and the dilution refrigerator are integrated in a common cryostat with an internal vacuum barrier between the insulating cryogenic vacuum and the very high beam vacuum. In addition, the magnet system has to guarantee a smooth transition between the 5 T and the 1 T magnetic field areas required for a loss-free transfer of antiprotons and positrons from the trapping region to the antihydrogen production area. In this paper the design of this AEGIS magnet system is presented and discussed.