The monarch is experiencing "mild cold-like symptoms" but expects to continue "light duties" at Windsor over the coming week, the palace said.
"She will continue to receive medical attention and will follow all the appropriate guidelines," it added in a statement.
Her eldest son and heir, the Prince of Wales, tested positive on 10 February, after meeting his mother on 8 February.
It is understood a number of people have tested positive at Windsor Castle, where the Queen resides.
Later on Sunday, the Palace issued a congratulatory message from the Queen to the Team GB Curling teams.
She heralded the teams' "outstanding performance" at the Winter Olympics, which led to the women securing a gold medal and the men silver.
She added: "I know that your local communities and people throughout the United Kingdom will join me in sending our good wishes to you, your coaches and the friends and family who have supported you in your great success."
Prime Minister Boris Johnson said he was "sure I speak for everyone in wishing Her Majesty The Queen a swift recovery from Covid and a rapid return to vibrant good health".
Leader of the Opposition, Labour's Sir Keir Starmer, also wished the monarch "good health and a speedy recovery", adding: "Get well soon, Ma'am."
The positive test comes days before England is expected to drop the legal requirement to self-isolate for those who catch Covid, as the last virus restrictions are set to be removed.
Outlining the thinking behind the government's upcoming "living with Covid" plans, Mr Johnson told the BBC it was time to "shift the balance away from state mandation" and said the public can "get their confidence back".
On the eve of her Platinum Jubilee marking 70 years on the throne on 6 February, the Queen carried out her first major public engagement for more than three months, meeting charity workers at Sandringham House.
The Queen, who will be 96 in April, had her first vaccine in January 2021 and is believed to have had all her follow-up jabs after that.
BBC royal correspondent Nicholas Witchell said she had been taking life "rather more easily" since spending a night in hospital for medical checks in October last year.
In November, the Queen missed the Remembrance Sunday service with a strained back, weeks after being advised to rest by doctors following the hospital stay.
It was the first time she missed the ceremony at the Cenotaph in London as monarch, other than when she was pregnant or abroad.
While on light duties it is likely the Queen will be reading documents from UK government ministers and Commonwealth representatives sent to her every day, approving and signing them where necessary.