According to preliminary information, the woman in her thirties and her husband were staying with their two children, the baby and a-three-year old, at a holiday home in Saint-Pee-sur-Nivelle, in the Bayonne area.
Saturday was the last day of their week-long holiday and the father woke up early and left the bedroom which all four were apparently sharing, according to the official from the Bayonne prosecutor's office.
The mother only got up later, around 9 am (0700 GMT) and it was then that the father discovered that the baby was dead and alerted the emergency services.
"The suspect death of a baby was discovered at the scene" and police enquiries quickly focussed on the mother, who was arrested, deputy prosecutor Marc Mariee told AFP.
A post-mortem will be carried out next week.