The 44-year-old former Olympics champion, who married Prince Albert in 2011, returned to Monaco in March as her health improved, the royal palace said at the time. She only recently resumed official duties.
The palace said there were no concerns about her condition and she would isolate according to health protocols.
The positive test comes a day after France's Public Health Authority said it had seen a rebound in cases this week for the first time in months with a 7-day average of about 20,000 daily cases.
Guillaume Spaccaferri, epidemiologist at the Public Health Authority, told a news conference on Friday the rise in numbers was likely down to a new Omicron variant and the end of all Covid-19 restrictions in the country.
So far, there was not a notable impact on hospitalisations or intensive care units, he said.
(Reporting by John Irish; Editing by Mark Potter)

Princess Charlene of Monaco arrives for the start of the Formula One Grand Prix of Monaco at the Circuit de Monaco in Monte Carlo, Monaco, 29 May 2022. EPA-EFE/CHRISTIAN BRUNA