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Boris Johnson says he spoke to the Queen this week and she was on "very good form", adding the whole country "wishes her well".
It comes after the 95-year-old monarch was advised by doctors to rest for at least the next fortnight.
Her Majesty accepted their recommendations that she only takes on "light, desk-based duties" and not to carry out any official visits, Buckingham Palace said on Friday.
The decision means she will be unable to attend the Festival of Remembrance at London's Royal Albert Hall on Saturday 13 November.
However, the palace said she has the "firm intention" of leading the nation in honouring the country's war dead on Remembrance Sunday the following day.
It is significant that the monarch wishes to attend the ceremony at the Cenotaph on Whitehall, as it is a major event on the sovereign's calendar and one to which she has attached great importance.
The prime minister said on Saturday: "I spoke to Her Majesty, as I do every week, this week, and she's on very good form.
"She's just got to follow the advice of her doctors and get some rest and I think that's the important thing. I'm sure the whole country wishes her well."
On Thursday, Buckingham Palace shared footage of Her Majesty presenting David Constantine with The Queen's Gold Medal for Poetry.
In the short clip, the Queen jokes about having to speak to him via video-link saying: "I'm very glad to have the chance to see you, if only mechanically this morning!"
Earlier this week it was announced she had pulled out of attending, in person, the upcoming COP26 climate summit which begins in Glasgow this weekend, after a busy recent schedule.
The Queen spent Wednesday night last week in hospital after cancelling a visit to Northern Ireland.
She was admitted for "preliminary investigations" but returned to Windsor Castle the following day.
It has not been revealed what medical tests she had at the private King Edward VII Hospital in central London. It was her first overnight hospital stay in eight years.
Sky's royal correspondent Rhiannon Mills said Her Majesty's hospital admission was not related to COVID-19.
In recent days, the Queen carried out her first official engagements since her brief hospital stay as she held virtual audiences at Windsor Castle.
She wore a yellow dress and her three-strand pearl necklace as she met the South Korean ambassador, Gunn Kim and, separately, the Swiss ambassador, Markus Leitner, on Tuesday.