Departing Ipswich captain Morsy led the club to back-to-back promotions and into the Premier League last season.
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The Egypt defender, 33, leaves for Kuwait SC in a free transfer.
"We have had some incredible days and nights together and I will remember many of these for the rest of my life," he told the club website.
"Achieving back-to-back promotions allowed me and many of my team-mates to achieve our dreams of playing in the Premier League, so I will always feel indebted to (boss) Kieran (McKenna) his staff, everyone at the club and the supporters for helping us get there."