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The Alpha Magnetic Spectrometer (AMS) is a particle detector designed to search for anti-matter, dark matter and the origin of cosmic rays in space. The detector will be assembled at ETH Zurich and installed on the International Space Station (ISS) in 2004. The planned duration of the experiment is 3 years. The magnetic dipole field is achieved by an arrangement of 14 superconducting coils. The magnet system consists of a pair of large Helmholtz coils together with two series of six racetrack coils, circumferentially distributed between them. This arrangement was mainly chosen to minimize the stray field outside of the magnet and to minimize the magnetic dipole moment. It generates a magnetic field of 0.87 T in the center of the magnet with a bending power of 0.78 Tm2. All superconducting coils are wound from a high purity aluminum-stabilized mono-strand NbTi conductor with rectangular shape of 1.55 mm × 2.00 mm. The coils are located inside a toroidal-shaped vacuum vessel. They are indirectly cooled by superfluid helium at 1.8 K. This cooling loop is thermally connected with a 2500 l vessel for superfluid helium which serves as a cold reservoir. In order to ensure the 3 year endurance without refilling, the magnet design was optimized with respect to very low heat losses. This paper describes the main features of the AMS superconducting magnet including the manufacturing process of the conductor and the principle concept of the cryogenic system.