Meghan Markle has demanded the Mail on Sunday pay her $2.7 million in legal fees and hand over copies of her letter to her estranged father.
Last month a High Court judge ruled that the publisher had breached Meghan's privacy after printing parts of the letter she wrote to Thomas Markle, reports the Sun.
In a virtual hearing this week, Meghan's lawyers asked the High Court to order Associated Newspapers to hand over any copies of the letter and destroy any electronic copies made, as well as any notes made about it.
Meghan also said her legal costs were around $2.7m, and demanded 50 per cent to be paid within the next two weeks.
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Lord Justice Warby ordered the Mail on Sunday to pay her $800,000 in costs, with more to be paid in future.
Ian Mill QC, who represented Meghan, told the court Meghan was willing to "cap her damages" for misusing private information "at a nominal award" to "avoid the need for time and cost to be incurred in debating these issues".
The duchess has also asked that the Mail on Sunday print a front page statement saying she had won her privacy case, and that the Mail Online have a piece on their website's homepage for six months.
Antony White QC, who represented Associated Newspapers, said printing the statement was unnecessary and seemed to be "intended more as a species of punishment or retribution, rather than as a necessary and proportionate measure in the interests of the claimant or the public".
This week Lord Justice Warby refused to order the delivery or destruction of any copies of Meghan's letter to her father "at this stage".
Details of the order for publication would be provided in a written ruling handed down at a later date.
The Mail on Sunday's application to appeal the privacy and copyright decision was refused. It will, however, be able to apply to the Court of Appeal.