Brown, who also played for Rangers, replaces Barry Smith, who was sacked on Wednesday with the club 15 points adrift at the bottom of the Clydesdale Bank Premier League.
The club announced the news with a statement on their official website a day ahead of Sunday's visit to Celtic Park.
It read: "Having been inundated by high calibre candidates over the last few days, one man stood out as perfect for the position of interim manager of Dundee FC.
"John Brown is a man with real heritage and history at Dundee FC and when at Dens Park in his time as a player, he was an real inspiration, a fans' favourite and scored some vital goals and he has never lost his feelings for the club.
"His time at Rangers was highly successful with six league titles numbered amongst a glittering career and once more he was an inspirational leader. He has been a coach at Rangers and Clyde and was a key member of the Ibrox coaching and scouting set up until last summer.
"Having met and spoken with members of the board and CEO Scot Gardiner this week, his passion for Dundee was instantly evident and his willingness to step up to the plate for the remainder of the season alongside his immediate understanding of what we as a club need from the position, convinced us he is the man to address the challenges ahead of us this season.
"There is still a lot to play for this season with some vital games coming up and John we will make sure that the Dundee fans see and understand that he is here to make a difference and he is looking forward to working with Assistant Manager Ray Farningham, the coaching staff and our full squad of players.
"The club will continue to assess and review the longer term situation but as has we have highlighted, due to the mixed challenges of our current league position and the uncertainties over league reconstruction, it is impossible for us to set a budget for next season and coming years. Hopefully that position will become clearer very soon and we will then look to appoint a manager in a full time, longer term position.
"In the meantime we hope that every Dundee fan will rally behind John and the team for the remainder of what can still be an exciting and historic season."
Brown is a surprise appointment given his lack of recent managerial experience.
The former Rangers defender's only managerial job, at Clyde, ended in November 2009 after less than two years in the post.
But he had two coaching spells with Rangers and left his role as a scout when Charles Green took control at Ibrox in June. Brown then launched a vociferous attempt to lead a fans' buyout of Green's consortium.
Dundee hope Brown, who spent four years at Dens Park as a player, can prove to be the "inspirational leader" he was when helping Rangers to six league titles.
Although relegation looks a near certainty, Dundee have a huge match a week tomorrow when they face city rivals United in the William Hill Scottish Cup quarter-finals.
Dundee face SPL leaders Celtic at Parkhead tomorrow but it is unclear how hands on Brown will be after Farningham prepared the team in the wake of Smith's sacking.
But they will have one new face in their ranks after completing the signing of former Shamrock Rovers striker Don Cowan, who Smith brought in as a trialist after he was released following an injury-hit spell with Stevenage.
Farningham has been tasked with lifting the players following the departure of the popular Smith, who led the club through administration and into the SPL.
Farningham told the club's official website: "It couldn't be harder facing the league champions, but all the pressure is on Celtic. There's no pressure on the boys, I said that today, they are not expected to take anything from the game.
"Go there relax and just play. A lot has gone on this week, (they have) just got to clear their minds and get focused on the game.
"They've been lively. Obviously I had to tell a sad bunch of players what actually happened and things like that but they've responded well and they've been very professional, which you have to be.
"I feel for Barry as well but the players, as I say, have responded positively."
Dundee have defender Davide Grassi and striker Mark Stewart back from injury but defender Lewis Toshney cannot play under the terms of his loan agreement with Celtic.
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