All 27 people aboard the Grande America were rescued without injury on Monday night.
The vessel was travelling from Hamburg, in Germany, to Casablanca, in Morocco, before the blaze started.
The fire “intensified over the past 24 hours”, French naval authorities said on Twitter.
“The ship had been listing increasingly to the right as time went on,” the statement continued.
A first tugboat to reach the ship was unable to extinguish the fire, and two others were on their way to the site on Tuesday when the ship sank in waters that are around 4,600 metres deep.
According to Italian news agency ANSA, the ship was transporting “365 containers, 45 of them classified as containing dangerous matter”, and more than 2,200 tonnes of diesel.
French Environment Minister Francois de Rugy told lawmakers in Paris on Tuesday that anti-pollution measures were being taken.