CMS (compact muon solenoid) is a general-purpose detector designed to run at the highest luminosity at the CERN Large Hadron Collider (LHC). Its distinctive features include a 4 T superconducting solenoid with 6 m diameter by 12.5 m long free bore, enclosed inside a 10,000-ton return yoke. The stored magnetic energy is 2.6 GJ. The magnet is being assembled in a surface hall and will be tested at the beginning of 2005 before being transferred to an experimental hall 90 m below ground level. The design and construction of the magnet is a common project of the CMS Collaboration. The task is organized by a CERN based group with strong technical and contractual participation of CEA Saclay, ETH Zurich, Fermilab, INFN Genova, ITEP Moscow, University of Wisconsin and CERN. The return yoke, 21 m long and 14 m in diameter, is equivalent to a thickness of 1.5 m of saturated iron interleaved with four muon stations. Manufacture of the yoke and vacuum tank is completed and the first sub-detectors have been installed. The indirectly-cooled, pure-aluminum-stabilized coil is made up from five modules internally wound with four layers of a 20 kA mechanically-reinforced conductor. The manufacture of the conductor is completed and winding is in progress for a final assembly in 2004. All ancillaries are delivered or under contract. The magnet project is described, with emphasis on the present status of the fabrication.