The Duke of Sussex and former Labour deputy leader Lord Tom Watson are suing News Group Newspapers (NGN) over allegations of unlawful information-gathering.
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 though – who said the two sides had already had “ample time to seek to resolve their differences”.
Both sides 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.”
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.”