The young woman died in hospital in the city of Poitiers, in the Vienne department, the regional health authority (ARS) in Bordeaux announced on Tuesday.
According to hospital authorities the 32-year-old woman had not been vaccinated against the virus. She was admitted to hospital on February 1st and quickly transferred to intensive care but not did survive.
The Nouvelle-Acquitaine region of western and south western France (see map below) has been hit by an epidemic of measles since November which has resulted in 269 confirmed cases, of whom 66 were admitted to hospital for treatment. Around one quarter of those admitted to hospital were treated in intensive care.
Those figures represent a jump by around 50 percent in the number of measles cases since January 24th, the last time authorities released data.
French health authorities said the death of the woman underlined the importance of being vaccinated against measles and asked the public to check their medical records to make sure they were.
"It's the only way to stop the epidemic," they said reminding the public that "there is no cure for the disease" which is highly contagious.
"Measles is not a benign disease, especially for children and can lead to respiratory and neurological complications that can have serious consequences," said the regional health authority (ARS) in a statement.
The hospital in Poitiers believes at least five people hospitalized there contracted the virus by coming into contact with another patient.
Hospital chiefs have no enforced a rule that all members of the public entering the emergency wards must wear a mask. Some 54 members of staff have had to be vaccinated.
Authorities are worried that the rate of vaccinations in the Nouvelle-Acquitane region is too low to cope with the epidemic. It is believed between 70 and 81 percent of the public are vaccinated but the level recommended by the World Health Organisation is 95 percent.
In France, children are vaccinated against measles, mumps and rubella when aged between 12 and 18 months.