The 24-year-old joins the Tannadice club after his contract with Dunfermline Athletic expired. Morris revealed talks with United manager Jackie McNamara were crucial to his decision.
He said: "After speaking to the manager I was set on joining United. This is a great club which showed last season that it is capable of producing one of the best teams in Scotland and I want to play my part in improving them further.
"Dunfermline were great for my career and I now want to continue my progress and coming to Tannadice is a great move for me.
"There are many talented footballers here and I am looking forward to playing alongside them."
Meanwhile, Ryan Conroy last night left city rivals Dundee after spending three seasons at Dens Park.