Children of Pure-bloods and Half-bloods are Half-bloods
JKR has stated on her old website that this was the case, however was quick to establish that this would only be of importance to those who feel the need to prejudice to that extent
The expressions 'pure-blood', 'half-blood' and 'Muggle-born' have been coined by people to whom these distinctions matter, and express their originators' prejudices. As far as somebody like Lucius Malfoy is concerned, for instance, a Muggle-born is as 'bad' as a Muggle. Therefore Harry would be considered only 'half' wizard, because of his mother's grandparents.
FAQ at J. K. Rowling's Official Site
This is re-iterated by Bellatrix Lestrange in the fifth book.
“Shut your mouth!” Bellatrix shrieked. “You dare speak his name
with your unworthy lips, you dare besmirch it with your half-blood’s
tongue, you dare —”
“He dared — he dares —” shrieked Bellatrix incoherently. “— He
stands there — filthy half-blood —”
Harry Potter and the Order of the Phoenix, Chapter 35 (Beyond the Veil)
There was, however, precedence for pure-blood families to allow half-bloods to marry into their bloodlines:
From the imposition of the Statute of Secrecy onwards, no Malfoy has married a Muggle or Muggle-born. The family has, however, eschewed the somewhat dangerous practice of inter-marrying within such a small pool of pure-bloods that they become enfeebled or unstable, unlike a small minority of fanatic families such as the Gaunts and Lestranges, and many a half-blood appears on the Malfoy family tree.
The Malfoy family - Writing by J.K. Rowling
Although there remained those families that married only within pure-blood circles, such as the Gaunts and the Lestranges, listed above.
Centuries it’s been in our family, that’s how far back we go, and pure-blood all the way!
Harry Potter and the Half-Blood Prince, Chapter 10 (The House of Gaunt)
It is worth noting that although most Death Eaters were half-blood, they would've pretended to be pure-blooded and resented their muggle sides. (Such as Voldemort and Snape did)
"“The Death Eaters can’t all be pure-blood, there aren’t enough pure-blood wizards left,” said Hermione stubbornly. “I expect most of them are half-bloods pretending to be pure. It’s only Muggle-borns they hate, they’d be quite happy to let you and Ron join up.”
Harry Potter and the Half-Blood Prince, Chapter 12 (Silver and Opals)