Who wrote “Twinkle, Twinkle Little Star?”

The popular story is that 6-year-old prodigy Wolfgang Amadeus Mozart wrote that little ditty. He wrote variations on it in the 1780s, which were probably exercises for his music students. They were described as being based on a French folk melody. It was popularized as a song about candy: Ah! Vous Dirai-Je, Maman. So who actually wrote the melody? Unknown. It is thought to have been composed in the 1740s. (Mozart was born in 1756.)