The case, made in 1514 and known as a masterpiece example of 16th century metal work, had been on display in a special glass container in the museum since Tuesday April 10. It has been property of the museum since 1886.
It was stolen overnight from Friday April 13 to Saturday April 14.
It was one of three exhibits to be stolen: a golden Hindu statue, and a collection of golden coins also disappeared.
A police inquiry has now been opened into the theft.
Department president Philippe Grosvalet released a statement condemning the actions of the robbers.
He said: “These burglars have attacked our common heritage, including this invaluable piece. Much more than merely a symbol, Anne de Bretagne’s heart belongs to our shared history. Queen Anne, Duchess of Brittany, had wished that her heart be buried with her parents. It was saved from melting after the Revolution.”
Nantes mayor Johanna Rolland said she was “indignant” over the theft, and called for it to be returned as soon as possible.
She said: “This magnificent object, a masterpiece of early 16th century metal work, naturally has great heritage value. For all the women and men of Nantes, it also has a special sentimental importance, linked to the personality of Anne de Bretagne and the imprint she left in the history of Nantes.
“I therefore hope that this precious and fragile object will be found as soon as possible.”