Leeds have confirmed that Dennis Wise will take over as manager.
Wise took over Swindon in May
The former England midfielder, 39, is joined at Elland Road by assistant Gus Poyet after the pair linked up successfully at former club Swindon.
A statement on the Leeds website said: "A compensation package has been agreed between the two clubs.
"Swindon Town have placed on record their thanks to Leeds United for the professional manner they have made their approaches."
Wise and Poyet will officially take up their posts on Wednesday, while goalkeeping coach Andrew Beasley has also joined the club's staff.
Wise's appointment has not been universally popular among Leeds fans, but club chairman Ken Bates, a close friend of Wise from their time at Chelsea, defended the move.
"After much deliberation the unanimous choice was Dennis Wise and his assistant Gus Poyet," he said.
"I am aware that he may not be everybody's cup of tea but this is not a popularity contest.
"It is a question of the right man to resolve a serious situation."
Previous manager Kevin Blackwell was sacked on 20 September, leaving John Carver to take over in a caretaker role.
However, Leeds have since slipped to second from bottom in the Championship after a run of four straight defeats.
Carver left the club on Monday as Leeds thrashed out personal terms with Wise.
Swindon, meanwhile, have asked club captain Ady Williams to take over as caretaker manager as they look for a successor to Wise, just six months after his arrival at the club.
Wise took the reins at the County Ground in May following Swindon's relegation to League Two and they won their first six league games.
However, a clause in his and Poyet's contracts allowed them to speak to Premiership and Championship clubs, provided suitable compensation was agreed.
A Swindon statement read: "Although we are all disappointed that the duo's tenure at the club turned out to be brief, the board would like to thank Dennis and Gus for steering the team to its current League Two position.
"We are confident that the squad they have assembled will continue to improve and generate further success for the club.
"Dennis and Gus were both offered improvements to their Swindon contracts, but they felt that the opportunity to manage at a higher level was the best move for both their careers and their families at this time."