"He's a Death Eater. He's replaced his father as a Death Eater! ... In Madam Malkin's. She didn't touch him, but he yelled and jerked his arm away from her when she went to roll up his sleeve. It was his left arm. He's been branded with the Dark Mark."
-- Harry, speculating while on the Hogwarts Express (OP7)
Harry is convinced that he has become a Death Eater, but we never learn if this is actually true. Hermione, for one, doesn’t believe it.
We can't say for sure when Draco would have been given the Dark Mark, but it would have been about the same time as Narcissa and Bellatrix's visit to Snape at the end of June, 1996. Certainly it was before the visit to Diagon Alley when Harry, Ron, and Hermione encountered him in Madam Malkin's, which was August 3. We'll place it as July, 1995.
There is a particularly compelling fan theory bouncing around the internet that Draco is not in fact a Death Eater but has been bitten by Fenrir Greyback as punishment to his father. Interesting idea, found here: http://www.dracomalfoyisawerewolf.com/
Sadly, Rowling shot this one down on Twitter.