[credit: original image via Metropolitan Museum of Art; colorized by me]
I came across this miniature via the online Met collection and decided to spend a few minutes colorizing it. When I saw the miniature I thought that it looked a bit like Madame Elisabeth and sure enough, according to the Met's "Signature, Inscription and Markings" information, this portrait is inscribed as depicting Madame Elisabeth, sister of Louis XVI. It is from the collection of the famous collector Bernard-Franck, a veteran of the 1870 Franco-Prussian War who was known for his important collection of paintings from the late 18th and early 19th century.
The miniature is undated but given the style, outfit and apparent age of Elisabeth (assuming the inscription is correct) then it is likely late 1780s.