“I would like you to know how very sorry I am for my part in the accident,” he wrote, on Sandringham House headed paper.
“The sun was shining low over the main road. In normal conditions I would have no difficulty in seeing traffic coming… but I can only imagine that I failed to see the car coming, and I am very contrite about the consequences.”
Ms Fairweather had previously criticised the duke for a lack of communication following the crash.
The mother-of-two told the Sunday Mirror: “I thought it was really nice that he signed off as ‘Philip’ and not the formal title. I was pleasantly surprised because of the personalised nature.”
After the crash, Royal biographer Hugo Vickers told BBC News: “Any kind of car accident at the age of 97 is likely to produce shock.
“Some years ago he gave up flying planes long before he needed to because he was scared that if something happened there would be a lot of criticism.
“You know, why was he, at the age of 55, still flying a plane when he should have retired at 48 or something like that.
“So he does listen to these things – he’s very, very sensible.”
In 2016, the duke famously drove the Obamas when the then US president and First Lady visited Windsor.
Prince Philip retired from public life in August 2017 having spent decades supporting the Queen and attending events for his own charities and organisations.
Buckingham Palace calculated he had completed 22,219 solo engagements since 1952.
Since retiring from official solo duties, he has appeared in public alongside the Queen and other members of the Royal Family at events and church services.
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