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As part of the collaboration program between CERN and KEK for the LHC, we have been developing a high field gradient superconducting insertion quadrupole magnet with a field gradient of 240 T/m in an aperture of 70 mm. To date two model magnets have been built and their quench characteristics and field qualities have been studied. Based on these results. We have performed the final adjustment of the magnet design. In this paper we present the summary of the design which includes short sample limits and field quality analysis for both the straight section and the end regions.