We know that Ice, the historical sword of House Stark was reforged after the events from the first book:
"...and had Ice melted down and reforged. There was enough metal for two new blades."
A Storm of Swords
By all accounts the art of forging/reforging Valyrian steel was lost after the Doom. How was Ice melted down and then reforged anew when the art of this would have been lost to time?
Tobho Mott was a smith from Qohor who knew the spells to rework the metal.
Tobho had learned to work Valyrian steel at the forges of Qohor as a boy. Only a man who knew the spells could take old weapons and forge them anew.
A Game of Thrones - Eddard VI
Ice was reforged by Mott. (at least it was in the books)
"Nor I, my lord," said the armorer. "I confess, these colors were not what I intended, and I do not know that I could duplicate them. Your lord father had asked for the crimson of your House, and it was that color I set out to infuse into the metal. But Valyrian steel is stubborn. These old swords remember, it is said, and they do not change easily. I worked half a hundred spells and brightened the red time and time again, but always the color would darken, as if the blade was drinking the sun from it. And some folds would not take the red at all, as you can see. If my lords of Lannister are displeased, I will of course try again, as many times as you should require, but—"
A Storm of Swords - Tyrion IV
"You have done good work, Master Mott," Lord Tywin told the armorer. "My steward will see to your payment. And remember, rubies for the scabbards."
A Storm of Swords - Tyrion IV
The secrets seem to be available only to Qohorik smiths.
The properties of Valyrian steel are well-known, and are the result of both folding iron many times to balance and remove impurities, and the use of spells—or at least arts we do not know—to give unnatural strength to the resulting steel. Those arts are now lost, though the smiths of Qohor claim to still know magics for reworking Valyrian steel without losing its strength or unsurpassed ability to hold an edge.
The World of Ice and Fire - Ancient History: Valyria’s Children
Only here, in all the world, has the art of reworking Valyrian steel been preserved, its secrets jealously guarded.
The World of Ice and Fire - The Free Cities: Qohor
No one's made a Valyrian steel sword since the Doom of Valyria. There are three living smiths who know how to rework Valyrian steel. The finest of them was in Volantis. Came here to King's Landing at my invitation.
In this episode, Tywin Lannister explains to Jaime that he found someone who knew how to forge and reforge the sword. It seems that art of forging/reforging Valyrian steel, is not lost. Its just rare.
commissioned a Volantis blacksmith, the best of those who can work with Valyrian steel, to re-forge it.
Maybe it's simply the art/science of manufacturing Valyrian steel that was lost, and a really good blacksmith can still work with it. I remember reading a piece of (perhaps dubiously canon) lore that said that the production of Valyrian steel involved the use of magic. That would really cut down on the number of people who can make Valyrian steel in the Seven Kingdoms, since there are very few people who could know enough about magic to do this (the various followers of the Lord of Light, and maybe a couple maesters) and none of them are common (or particularly likely to become professional blacksmiths).
This isn't a hard answer, though. I don't think it was ever explicitly stated how it was done, or that Valyrian steel was actually impossible to make, but my book knowledge isn't good enough to confirm or deny any of this.
