"My attention was first drawn to Scott when he was playing in the Dundee youth team against Hearts," said the former Tynecastle boss.
"A while later, I tried to sign him and I had another go when I was down in England, but nothing came of it.
"In the end, it's taken five or six years, but it will be worth it because we are getting another talented player whose best years are ahead of him and who has the ambition and hunger to be a success."
Robertson made his debut for Dundee in January 2005, going on to make more than 100 appearances for the club.
As skipper last season, he helped Dundee reach second place in the First Division, finishing seven points behind Hamilton.
"Scott is a very direct player who likes to get in the opposition box whenever possible," added Levein.
"I believe he will help us strengthen our goal threat from open play."
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