Pugh Auctions said the property had been partly cleared by the current owner, but renovations would be needed.
The building fetched £482,000 when it was first sold by the same auction firm.
The library site had been vacant since May 2022 and was deemed surplus to requirements by Leeds City Council.
The authority said the building was too big for the library service and alternative uses could not be found.
In January 2024, about 70 campaigners held a protest outside the building, saying they were “angry” at the choices Leeds City Council had made.
But speaking at the time, a council spokesperson said the authority had to sell the building due to its financial situation.