The ending is real but late — added roughly three centuries after the proverb, and the original was not about curiosity at all.
Two things tend to drop out of the way this one is usually told. The first: the proverb did not begin with curiosity. It began with care, meaning worry or sorrow, and it is first recorded in Ben Jonson's 1598 play Every Man in His Humour — care'll kill a cat. Shakespeare, whose company performed that play, used it the following year in Much Ado About Nothing, where a character is told that even if care killed a cat, he has spirit enough in him to kill care.
Care was still doing the killing as late as 1898, when Brewer's Dictionary of Phrase and Fable recorded it and noted drily that although a cat has nine lives, care would wear them all out. The curiosity version appears in the nineteenth century — an Irish newspaper in 1868 reports that they say curiosity killed a cat once — and gradually displaced the original.
The second correction concerns the famous ending. But satisfaction brought it back is genuine, but it is a much later rejoinder: a 1905 version has the cat merely coming back, and the wording with satisfaction is first recorded around 1912. So it is not the lost original half of an ancient saying. It is a good joke someone added three hundred years afterwards, and the internet has promoted it to scripture.