The CMS experiment (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 a free bore of 6 m diameter and 12.5-m length, enclosed inside a 10 000-ton return yoke. The magnet will be assembled and tested in a surface hall at Point 5 of the LHC at the beginning of 2004 before being transferred by heavy lifting means 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 from CEA Saclay, ETH Zurich, Fermilab, INFN Genova, ITEP Moscow, University of Wisconsin and CERN. The magnet project will be described, with emphasis on the present status of the fabrication.