Mr Astaire said: "The Princess is on the mend. She got it three weeks ago but she's getting better. She's over the worst.
"She suffered extreme fatigue and had regular fevers."
WATCH: Prince William congratulates Oxford researchers on vaccine breakthrough
He added on Monday: "I spoke to her today and she sounded much better."
Mr Astaire told The Sun on Sunday over the weekend: "Princess Michael of Kent’s housekeeper fell ill three weeks ago, and HRH was immediately tested and was found to be positive for COVID."
The Princess, 75, has remained in isolation at her home at Kensington Palace since the diagnosis. Her husband Prince Michael, 78, isolated for 14 days and tested negative for the disease.
The royal's housekeeper, who isolated after she developed mild symptoms, has since recovered.
Princess Michael with her husband Prince Michael of Kent
The news comes after it was revealed earlier this month that the Duke of Cambridge had tested positive for COVID-19 back in April.
The father-of-three reportedly kept his condition private as he didn't want to alarm the nation.
Prince William's father, Prince Charles, also contracted coronavirus in March after developing mild symptoms.
The Duke of Cambridge joined Oxford University researchers on a video call on Monday to congratulate them on their breakthrough with a vaccine that is shown to be effective at preventing COVID-19 and offers a high level of protection.
William said: "Well done, I'm so pleased for all of you, I really am.
"I saw it in everyone's faces back in June how much time and effort was going into this, and I could see that there was a lot of pressure on everyone, so I'm so thrilled that you've cracked it – so really well done."
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