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The design and construction of a wide-aperture, superconducting quadrupole magnet for the LHC insertion region is part of a study towards a luminosity upgrade of the LHC at CERN. The engineering design of components and tooling, the procurement, and the construction work presented in this paper includes innovative features such as more porous cable insulation, a new collar structure allowing horizontal assembly with a hydraulic collaring press, tuning shims for the adjustment of field quality, a fishbone like structure for the ground-plane insulation, and an improved quench-heater design. Rapid prototyping of coil-end spacers and trial-coil winding led to improved shapes, thus avoiding the need to impregnate the ends with epoxy resin, which would block the circulation of helium. The magnet construction follows established procedures for the curing and assembly of the coils, in order to match the workflow established in CERN's “large magnet facility.” This requirement led to the design and procurement of a hydraulic press allowing for both a vertical and a horizontal position of the coil-collar pack, as well as a collapsible assembly mandrel, which guarantees the pack's four-fold symmetry during collaring. The assembly process has been validated with the construction of two short models, instrumented with strain gauges and capacitive pressure transducers. This also determines the final parameters for coil curing and shim sizes.