Prince Rainier of Monaco's health remains "very fragile", his palace has said in a statement.
The prince has been in and out of hospital in recent months
The 81-year-old has been in hospital since 7 March and was taken into intensive care a week ago.
The palace said his condition had stabilised after three sessions of kidney dialysis.
But the state of his heart, lungs and kidneys meant the prognosis was "extremely guarded", the palace said.
"After three weeks of hospitalisation, after nearly eight days in intensive care, Prince Rainier remains in a very fragile state," Tuesday's statement said.
"... The heart, lung and kidney conditions [mean] the vital prognostic remains extremely guarded."
The prince's three children, Crown Prince Albert, 47, Princess Caroline, 48, and Princess Stephanie, 40, are taking turns at their father's bedside in hospital.
Rainier III, who acceded to the throne in 1949, is one of Europe's longest-reigning monarchs.
He made world headlines with his marriage to Hollywood star Grace Kelly in 1956 after meeting her at the Cannes film festival. The actress died in a car crash in 1982.
The prince's condition worsened on 22 March, when he was transferred to the hospital's intensive care.
Prince Albert is widely expected to succeed his father at the helm of the principality, which his family has ruled for more than 700 years.