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Precision muon momentum measurements at the highest LHC luminosity is a prime objective for the ATLAS experiment. To realise this objective the muon detector is based on a large, superconducting, air-cored toroid magnet system consisting of a long barrel and two end cap toroids. The end-cap toroids are required to produce strong bending powers, 4-8 Tm, over a radial span from 1.5-5 m. This paper presents the final engineering design which includes a number of design optimisation features. The optimisation of the cold mass structure to reduce internal stresses, the developments in resin technology to give greater safety margins in operation and novel techniques for cold mass support systems are described. Final design proposals for quench protection and control are presented with manufacture, assembly and installation plans.