As a result of her subsequent conviction in a British court, she now faces deportation from her Los Angeles home in September.
The US immigration office has told her that she must leave her three children, her husband and her job and return to England when her current humanitarian visa runs out.
"I just find it absurd that this is happening to me," she told the Independent on Sunday this weekend. The arrest and my treatment at the time were bad enough, but the fact that this incident is still threatening to ruin my life is just so terrible."
Now 42, Mrs Aaron works for a non-profit-making organisation which offers aid to the poorest members of the Jewish community across America, and her husband, David, works in a furniture store.
Her lifestyle since the drugs conviction, she argues, has been almost exemplary. She can see no reason why she should not be allowed to continue working in Los Angeles and living with her family, who are all American citizens.
"Since 1976, I think I have had a couple of speeding tickets and a parking ticket," she added wryly. "So, I suppose I am quite obnoxious really."
Mrs Aaron's lawyer, Jessica Croxton, who runs a small practice in Santa Monica, is battling to keep the family together.
"I plan to make a federal court challenge to prevent her deportation by making constitutional argument, and then I hope to persuade the British authorities that the case was mishandled in the first place," she said.
When 20-year-old Deborah Gabbay, as she then was, was first arrested, Ms Croxton believes she was denied important rights. She claims she was not permitted to make a phone call to a friend or a solicitor and she was kept in a cell for nine hours. "She appears to have had an initial confession coerced from her and when she did have legal advice this was not dealt with properly," said Ms Croxton.
"I wanted to plead not guilty," Mrs Aaron explained, "but my lawyer told me it would be better to go along with my initial confession that I had once smoked cannabis, and come away with a pounds 10 fine. That is what I did and that is what is now causing me such a problem."
If Mrs Aaron succeeds in persuading governments on either side of the Atlantic to alter their view, it will be the first time that the case law on cannabis convictions and US citizenship has changed since John Lennon secured his Green Card because of his exceptional status and because of the small amount of drugs involved in his British conviction.
Mrs Aaron's plight has come to a head now because of a new law introduced by the US government in April last year. The Illegal Immigration Reform and Immigrant Responsibility Act, known as IRAIRA, means that her humanitarian waiver is now only available in cases where it can be shown to be in the national interest for an immigrant with a criminal record to stay in America.Reuse content