J.K. Rowling confirms that Dumbledore knew the destroyed diary was the remains of a Horcrux;
JKR: "In essence divided," exactly, the "essence" being the soul. So Dumbledore knew all along that he must have- well, he suspected until Chamber of Secrets, and then at the point where he saw what was clearly the remains of a Horcrux, in other words, the diary, he thinks, "Okay. There you go. And not only has he definitely done it, but he's got to have more than one, because he's treated this one very casually."
[POTTERCAST - THE LEAKY CAULDRON - 12.23.07]
Yet Dumbledore allows Harry to take this very dark magical artifact and return it to Lucius Malfoy (in order to free Dobby):
‘Professor Dumbledore,’ he said hurriedly, ‘can I give that diary back to Mr Malfoy, please?’
‘Certainly, Harry,’ said Dumbledore calmly. ‘But hurry. The feast, remember.’
Harry grabbed the diary and dashed out of the office.
Chamber of Secrets - page 248 - Bloomsbury - chapter 18, Dobby's Reward
Why did Dumbledore allow Harry to take the Horcrux before he had fully examined it, especially as Dumbledore knew it was a Horcrux? The Peverell ring had a lethal curse placed on it for anyone who tried to put the ring on. Why didn't Dumbledore check the diary for additional curses or Dark Magic before giving it to Harry? I would have thought Dumbledore would have wanted to keep whatever Horcrux relics he found.
Also, do we know what happened to the diary after Harry gave it back to Lucius Malfoy and Lucius threw it aside? Are there any instances in canon that place the diary back in Dumbledore's hands?