Hans Kristian Rausing and his wife Eva had been due to appear at Westminster Magistrates’ Court in central London after she allegedly tried to take crack cocaine and heroin into the US embassy in London in April.
Police later searched their house in the city’s exclusive Chelsea district and are alleged to have found drugs there as well.
But now the charges against them will be shelved in favour of conditional cautions, prosecutor Martha Godwin told the court.
The couple’s lawyer Philip Smith called the decision “very sensible”.
Earlier, Hans Kristian Rausing’s family issued a statement saying they were “deeply saddened” by the events leading up to their planned court appearance.
“We hope with all our hearts that Hans K and Eva can overcome their addiction and we continue to do what we can to help,” it added.
The Rausing family, headed by Swedish billionaire Hans Rausing, is worth some £5.4 billion ($10.8 billion), according to this year’s Sunday Times Rich List.
Hans Kristian Rausing is Hans Rausing’s son.
The case was adjourned to August 12th, when the charges will be dropped formally. The couple, who were not in court, are on bail.