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The 62-year-old appeared to be the first head of state who has publicly said he contracted the virus.
In a statement, the palace shared Albert is being treated by doctors from the Princess Grace Hospital, named after his late mother. It revealed the royal is continuing to work from his home office in the palace and is in constant contact with the members of his government.
Albert urged residents of the small Mediterranean principality to respect confinement measures.
Albert is a five-time Olympic bobsledder and, in recent years, a global environmental campaigner.
The son of Prince Rainier of Monaco graduated from Amherst College in Massachusetts with a degree in political science in 1981 and is a member of the International Olympic Committee.
He and his wife, Charlene, Princess of Monaco, have twin children.
Kelly, a Hollywood star, became Princess Grace when she married Rainier in 1956. The Alfred Hitchcock muse passed away in 1982 at age 52 from injuries she sustained in a car crash.
For most people, the new coronavirus has caused only mild or moderate symptoms, such as fever and cough. For some others, especially older adults and people with existing health problems, it could cause more severe illness, including pneumonia.
The Associated Press contributed to this report.