The mother was on holiday with the father of the baby and her other child who is aged three. The woman, in her 30s, has not been named.
According to the Sud-Ouest newspaper, the mother indicated that she had smothered the child.
The family were on holiday at the Saint-Pee-sur-Nivelle region in the south-east of France. They were expected to return this weekend but have been held for further investigation.
The father of the child discovered that the child was unconscious in the morning. The emergency services were called and it was later discovered that the child had died.
The Bayonne prosecutor, Marc Mariée, told The Telegraph that, "The suspicious death of a baby was recorded at the scene."
He said the initial investigation "very quickly moved towards the mother, who has been placed in custody".
The custody period was extended on Saturday (3 September). An autopsy of the baby will be performed next week.