Bill Miller, whose company is based in the southern state of Tennessee, saw off competition from the Blue Knights, a consortium led by former Rangers director Paul Murray which enjoyed the backing of fans' groups.
The decision should mark the beginning of the end of one of the most turbulent episodes in the 140-year history of Rangers, who have won the Scottish title a record 54 times.
"We are delighted to announce that today we have received an unconditional bid for the business and assets of Rangers Football Club plc from Mr Bill Miller which has been accepted and he is now the preferred bidder," administrators Duff and Phelps said in a statement.
"Mr Miller now proposes to complete his transaction by the end of the season," it added. The season ends this month.
The deal has been structured in such a way that it avoids Rangers going into liquidation, a process that could have seen one of the biggest names in British soccer having to start life all over again at the bottom of the Scottish professional game.
Miller, an unknown quantity to Rangers fans, will have his work cut out to keep the club at the top in Scotland, where they and perennial "Old Firm" rivals Celtic have been dominant for decades.
Rangers went into administration more than nine million pounds in unpaid taxes and faced a much larger tax claim over how they paid their players over the past decade.
In addition, the Scottish football authorities have banned the club from signing players for 12 months and Premier League clubs could vote next week to impose additional point deductions on teams going into administration.
"Mr Miller and his team have sought clarity in relation to potential footballing sanctions and the place of Rangers Football Club plc within the Scottish Premier League," Duff and Phelps said.
"Significant progress has been made and discussions will continue throughout the period which Mr Miller now enjoys as preferred bidder."
