The duke's counsel, Stephen Suttle, told Mr Justice French at the High Court that there was no truth at all in the rumour.
The duke did not have a personal relationship with his butler and remained happily married to his second wife, Virginia. The couple wed in 1992.
The duke, whose seat is Floors Castle in the Scottish borders, brought the action over an article in the London Evening Standard last August.
Associated Newspapers Ltd publishers of the Evening Standard, and its former editor Stewart Steven acknowledged unreservedly that the rumour was wholly untrue and apologised for giving it further currency. They also agreed to pay the duke's costs in full.