The group, driven by former Wasps flanker Ken Moss, had been in talks for several months to take charge at Adams Park.
Contracts were recently exchanged, with the consortium purchasing shares held by previous Wasps owner Steve Hayes.
Wasps say they will continue to play at Adams Park in Wycombe "for the foreseeable future", but added that the new owners had placed a "long-term priority" on developing a new stadium.
The majority of Wasps office staff will be based at their Twyford Avenue training ground in Acton, with a new board of directors to be appointed shortly.
Former Wasps captain Mark Rigby will remain in his current role of club chairman, while David Thorne has already started work as the club's new chief executive.
A future role for Moss, whose work has been pivotal to the business and sport-based consortium assuming control, has yet to be finalised.
Rigby said: "We are delighted to confirm the completion of the sale process following the agreement of terms in recent months.
"It has undoubtedly been a lengthy process and there have been some complex financial issues involved, but we are pleased that the situation finally has been resolved.
"The vision of everyone involved in the consortium who will take the club forward is to ensure that London Wasps becomes a sustainable business long-term.
"In the short term we have some major work to get the company back on a sound financial footing, and our priority is to stabilise the business after a rocky and difficult 12 months for everyone involved.
"It is our hope to confirm our new board of directors, plus additional staff resource, as soon as possible.
"All stakeholders, including supporters, will be fully kept up to date with our progress and changes.
"We are also in the process of finalising a number of new sponsorship deals which, now we have confirmed ownership, will be progressed strongly."
Rigby also paid tribute to Hayes, adding: "I would like to thank Steve for his financial support over the last few years which shouldn't be underestimated.
"Under his ownership, the club sought to be innovative off the pitch with the development of the St George's Day brand and in taking the first English domestic competitive game overseas."
Wasps have lost their first two Premiership games of the season, but claimed three bonus points in the process. They host London Irish on Saturday.
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