As System Down stated, Ned returned the sword to Starfall after he was done with whole Tower of Joy business.
That cut deep. Ned would not speak of the mother, not so much as a
word, but a castle has no secrets, and Catelyn heard her maids
repeating tales they heard from the lips of her husband's soldiers.
They whispered of Ser Arthur Dayne, the Sword of the Morning,
deadliest of the seven knights of Aerys's Kingsguard, and of how their
young lord had slain him in single combat. And they told how
afterward Ned had carried Ser Arthur's sword back to the beautiful
young sister who awaited him in a castle called Starfall on the shores
of the Summer Sea. The Lady Ashara Dayne, tall and fair, with
haunting violet eyes. It had taken her a fortnight to marshal her
courage, but finally, in bed one night, Catelyn had asked her husband
the truth of it, asked him to his face.
AGOT - Catelyn II
George already confirmed in 2002 that the Sword is still at Starfall, waiting for another Sword of the Morning to rise and claim her. By customs of House Dayne, only the most puissant warrior of the family can wield the sword and be referred to as the Sword of the Morning.
Q: What happened to Ser Arthur Dayne's sword Dawn after Ned brought it
back to Ashara?
A: Dawn remains at Starfall, until another Sword of the Morning
So Spake Martin: VALYRIA-RELATED SUBJECTS (The Citadel)
The same guy asked about the Daynes and the Sword of the Morning,
asking how that title is decided. George said the Sword of the
Morning is always a member of House Dayne, someone who is deemed
worthy of wielding Dawn as decided within the House, that whoever it
is would have to earn the right to wield it.
So Spake Martin: US SIGNING TOUR (ALBUQUERQUE, NM) (The Citadel)