The baby died of a cardiac arrest on 21 December, France's official medicines watchdog ANSM announced on Monday, confirming a report in Le Figaro newsaper.
Investigations are under way to establish the exact causes of the death, the ANSM said.
The supplement, Uvésterol D, has been produced by the Crinex laboratory since 1990 and is widely distributed for Vitamin D deficiencies.
Although the ANSM stresses that no deaths have been attributed to it, it has been monitoring it since 2006 following reports of illness in babies who have been given it.The agencypublished precautions for its use in 2013, asking Crinex to "take measures to minimise risks", especially to babies and toddlers under five-years-old.
The independent review Prescrire has on several occasions warned that the method of administration of Uvésterol D, by pipette, is dangerous.