The Duke of Sussex and former Labour deputy leader Lord Tom Watson are suing News Group Newspapers (NGN) over allegations of unlawful information-gathering.
The Rupert Murdoch-owned NGN, which also ran the now-defunct News Of The World, denies any unlawful activity took place at The Sun.
The trial had been due to start at the High Court in London at 10.30am today but was delayed twice, before barristers for both sides requested an adjournment until 10am on Wednesday.
The request was refused by Mr Justice Fancourt – who said the two sides had already had “ample time to seek to resolve their differences”.
They could now go to the Court of Appeal in a bid to challenge the judge’s decision.
Requesting the third adjournment, NGN’s barrister Anthony Hudson said both parties “have been involved in very intense negotiations over the last few days and the reality is we are very close”.
He said: “Very unusually, both parties are in complete agreement that this is a very important step.
“The number of times the parties have been in agreement in this litigation are very few and far between.
“Both parties feel they have no choice but to persist in this.”
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Mr Hudson added there had been difficulties today due to “time difference issues” in reference to Harry, who lives in California.
David Sherborne, for the duke and Lord Watson, said: “Mr Hudson and I would not be asking for further time if we did not think it stood any prospect of potentially saving a lot of court time.”
Dismissing their request, the judge said: “I am not satisfied on the basis of what I have been shown so far that the trial should not proceed at this time and will therefore not allow the further adjournment.”
He added: “I’m not going to stand in the way of access to justice if the parties wish to go to the Court of Appeal.”
The trial is due to last 10 weeks and is scheduled to consider specific claims brought by Harry and Lord Watson, as well as “generic” allegations of wrongdoing – all of which NGN has vehemently denied.
Harry first brought his lawsuit in 2019. He has said he hopes to get the truth, after other claimants – including celebrities and politicians – settled to avoid risking multimillion-pound legal bills, which they could have been forced to pay even if they won in court.
In English law, if a claimant turns down an offer to settle outside of court, then even if they win, they are liable for the legal costs of both sides if the damages awarded are smaller than the settlement offer.