In the 2007/2008 season, Grant guided Chelsea to second place in the Premier League and also to the Champions League final, where the Blues lost on penalties to Manchester United and which turned out to be his last game in charge.
Portsmouth are currently bottom of the Premier League, with only seven points from their 13 games so far.
"Avram is a very experienced and respected manager who has managed at the highest level," added Storrie.
"The board believes he is the man to help steer the club out of the relegation zone.
"He will take charge of training on Friday alongside first-team coaches Paul Groves and Ian Woan."
Pompey executive director Mark Jacob said: "The decision's been made by the board and we're moving forward now.
"Everyone's very positive about us staving off relegation. He (Avram) knows the club intimately."
Hart was offered the chance to return to his old post of working in youth development, but he declined.
"I would like at this time to thank all of the supporters and players for the backing they have given during my time at Portsmouth Football Club," Hart said in a statement.
"The speed of recent events, though, has been somewhat surprising and I am naturally disappointed not to be able to continue in the role (of manager) at Portsmouth Football Club.
"Unfortunately I do not believe that having been the manager of Portsmouth Football Club I will be able to return to the youth-team role that has been offered to me by the club and it is with deep regret that my time with Portsmouth Football Club has now come to an end.
"However, I look forward to a return to management in the near future."
Hart, who was director of youth operations before replacing the sacked Tony Adams as manager in February 2009, was in charge during a particularly turbulent part of Portsmouth's history.
He kept them up last season but lost several high-profile players such as Peter Crouch and Glen Johnson over the summer, and this season the club's ownership has been in doubt.
Arab businessman Sulaiman Al Fahim bought the club from Alexander Gaydamak in August, but failed to come up with fresh investment, leading to a situation where players' wages were not paid on time in September.
In October Al Fahim sold his majority shareholding in the club to Saudi tycoon Ali Al Faraj, but any hopes Hart may have had of this making his job easier were hit when the Premier League imposed a transfer embargo on the club over debts owed to other clubs.
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