What happened to Ice, Eddard Stark's Valyrian steel sword?

Question

Ser Ilyn bowed before the king and queen, reached back over his shoulder, and drew forth six feet of ornate silver bright with runes. He knelt to offer the huge blade to Joffrey, hilt first; points of red fire winked from ruby eyes on the pommel, a chunk of dragonglass carved in the shape of a grinning skull.

Sansa stirred in her seat. “What sword is that?”

Tyrion’s eyes still stung from the wine. He blinked and looked again. Ser Ilyn’s greatsword was as long and wide as Ice, but it was too silvery-bright; Valyrian steel had a darkness to it, a smokiness in its soul. Sansa clutched his arm. “What has Ser Ilyn done with my father’s sword?”

I should have sent Ice back to Robb Stark, Tyrion thought. He glanced at his father, but Lord Tywin was watching the king.

Does this paragraph from A Storm of Swords confirm that Tywin Lannister reforged Ice into the two Valyrian steel swords meant for Joffrey and Jaime?

Answer 1

As far as we can tell, Ice was indeed reforged into two Lannister swords: Widow's Wail given to Joffrey, and Oathkeeper given to Jaime. So far in the story (Book 5), we have had no reason to doubt this story. Tywin had very good reasons to do what he did: he forever took away a priceless heirloom from the Starks, and he restored the Lannister status among the great houses who own Valyrian steel weapons, since the ancestral Lannister Valyrian steel sword Brightroar was lost a long time ago when King Tommen II Lannister took it with him to Valyria where he disappeared.

More on the fate of the new Lannister Valyrian swords:

Widow's Wail was passed on to Tommen after Joffrey's death, and Oathkeeper was gifted by Jaime to Brienne of Tarth.

Answer 2

This is made explicit by Jaime Lannister when he gives Oathkeeper to Brienne of Tarth and he tells her that Tywin Lannister had Ice reforged into two swords.

"Nor I. There was a time that I would have given my right hand to wield a sword like that. Now it appears I have, so the blade is wasted on me. Take it." Before she could think to refuse, he went on. "A sword so fine must bear a name. It would please me if you would call this one Oathkeeper. One more thing. The blade comes with a price."
...
"I know what you thought." Suddenly Jaime was sick of the sight of her. She bleats like a bloody sheep. "When Ned Stark died, his greatsword was given to the King's Justice," he told her. "But my father felt that such a fine blade was wasted on a mere headsman. He gave Ser Ilyn a new sword, and had Ice melted down and reforged. There was enough metal for two new blades. You're holding one. So you'll be defending Ned Stark's daughter with Ned Stark's own steel, if that makes any difference to you."
_{A Storm of Swords, Jaime IX}

The above also hints at how the passage in the OP Sansa Stark knew that something had happened to Ice as Ser Illyn Payne should have had it.

The other sword is Widow's Wail that was given to Joffery Baratheon.

The guests were shouting out names for the new blade. Joff dismissed a dozen before he heard one he liked. "Widow's Wail!" he cried. "Yes! It shall make many a widow, too!" He slashed again. "And when I face my uncle Stannis it will break his magic sword clean in two." Joff tried a downcut, forcing Ser Balon Swann to take a hasty step backward. Laughter rang through the hall at the look on Ser Balon's face.
"Have a care, Your Grace," Ser Addam Marbrand warned the king. "Valyrian steel is perilously sharp."
_{A Storm of Swords, Sansa IV}