The 38-year-old Norwegian has been appointed on a 12-month rolling contract. He will be officially unveiled as the successor to Neil Lennon at a press conference at Parkhead this afternoon.
After hard-fought negotiations, Celtic have agreed a compensation package with Stromsgodset, where Deila was under contract until the end of 2016.
The man who, in November, led the tiny, provincial club to their first championship for 42 years, will begin working in his new job immediately.
Deila said: "It is a magnificent honour to be named the new Manager of Celtic, one of the world's great football clubs and a club I have such enormous respect for. I will treasure this opportunity and will give everything I have to achieve success for Celtic and our wonderful fans.
"I know what I want for Celtic and our supporters and that is the best of everything, it is what our fans deserve. I want to deliver the best attacking, exciting and entertaining football we can play, for the players to give their best every time they take to the field of and of course I want my players to work with me to achieve the best results possible and bring trophies to Celtic.
"Celtic is in great shape and the Club has enjoyed real success in recent years due to the hard work of Neil and his team. I want to carry on this work and continue to bring happy times to Celtic. I can't wait to get started on this journey."
The Celtic players are due back for pre-season training on June 24, and will head to their training camp in Austria four days later.
After a handful of already-arranged friendlies, their first competitive game under Deila will be the first leg of the second qualifying round for the Champions League, which takes place on July 15 or 16.
Ironically, Stromsgodset will also enter the competition at this stage, and, as an unseeded side, could even be drawn against Celtic.
Deila's first move will be to decide who he wants as his assistant as Lennon's team of Johan Mjallby and Garry Parker have also left the club and coach, Danny McGrain, is recovering from a heart attack.
John Collins is understood to be the man under consideration, and the former Hoops player and manager of Hibs can provide Deila with a solid knowledge base of the football scene in this country, as well as a list of players who might be of interest to the new boss.
Deila already knows all about Stefan Johansen, who he sold to Celtic in January and who has proved to be an impressive acquisition to the Hoops side which just won its third title in a row.
Deila has a reputation for identifying young talent and developing players, which fits perfectly with the business plan the Parkhead club have been following for the past few years.
His teams play fast, attacking football, and this is one of the main reasons Celtic decided to look outwith the list of high-profile candidates who were believed to be leading the race to take over from Lennon.
Celtic Chairman Ian Bankier said: "We are delighted to have secured Ronny as Celtic manager. He is a highly regarded individual and a man who we are confident will bring great success to Celtic. He is a forward thinking coach and manager with a progressive approach. The Celtic Board will support Ronny fully and in every way and we look forward to working with him."
Celtic Chief Executive Peter Lawwell added: "We believe Ronny will prove to be a fresh and dynamic new manager and someone who will give us a bright and energetic way forward for Celtic.
"Knowing that he was so highly rated across Europe, Ronny was one of the first candidates we considered as manager and having discussed the opportunity with him at length, we are convinced he has the talent and the determination to deliver very positive results for the Club and our supporters.
"Ronny likes to play attacking, winning football, the Celtic way, something I know our supporters will endorse. As the new season begins, I am sure our fans will get right behind Ronny and the players once again as we aim to bring success to Celtic and continue to deliver a Club they can be proud of."
Henrik Larsson and Roy Keane ruled themselves out of contention, opening the door for Deila to emerge from left field and pip other very strong candidates, including Malky Mackay, Owen Coyle, Steve Clarke and Michael Laudrup, to the biggest job - and challenge - in Scottish football.