Dumbledore destroyed the Horcrux element of Marvolo Gaunt's Peverell ring with the Sword of Gryffindor, splitting the Resurrection Stone portion of the ring in half in the process.
Harry noticed a ring on [Dumbledore's] uninjured hand that he had never seen Dumbledore wear before: it was large, rather clumsily made of what looked like gold, and was set with a heavy black stone that had cracked down the middle.
Half-Blood Prince - pages 68-69 - UK Hardcover - chapter 4, Horace Slughorn
‘You destroyed the diary and I the ring, but if we are right in our theory of a seven-part soul, four Horcruxes remain.’ [Dumbledore to Harry]
Half-Blood Prince - page 475 - UK Hardcover - chapter 23, Horcruxes
‘I believe that the last time I saw the sword of Gryffindor leave its case was when Professor Dumbledore used it to break open a ring.’ [Phineas Nigellus to Hermione]
Deathly Hallows - page 250 - UK Hardcover - chapter 15, The Goblin's Revenge
When Ron destroyed the locket Horcrux with the Sword of Gryffindor, the locket was pretty much obliterated.
[There] was only Ron, standing there with the sword held slackly in his hand, looking down at the shattered remains of the locket on the flat rock. [...] Harry stooped, pretending he had not seen, and picked up the broken Horcrux. Ron had pierced the glass in both windows: Riddle’s eyes were gone, and the stained silk lining of the locket was smoking slightly.
Deathly Hallows - page 307 - UK Hardcover - chapter 19, The Silver Doe
When Harry puts two and two together and figures out the Resurrection Stone comes from Marvolo Gaunt's ring, and is hidden in the Snitch Dumbledore left him, Ron questions whether the stone would work properly.
Ron’s mouth fell open. ‘Blimey – but would it still work if Dumbledore broke –’
Deathly Hallows - page 348 - UK Hardcover - chapter 22, The Deathly Hallows
How does the Resurrection Stone survive the destruction of the Horcrux inside it, and how is it the stone works properly after being cracked in half by the Sword of Gryffindor (which is imbued with Basilisk venom)? Shouldn't being struck by the Basilisk venom-infused Sword of Gryffindor, and being exposed to the evil of Voldemort's soul as the Horcrux is destroyed, alter the Resurrection Stone's magical properties? I don't understand how it wasn't destroyed or rendered unusable.
☆ Sorry for the citations dump. If I don't include citations, typically someone asks me for them (which is totally fine!) So I went ahead and included them pre-emptively.