Saying it intended to be a long-term shareholder, Qatar said it saw itself "as a shareholder that will provide stability and support for the board's strategy of developing further its business".
The London Stock Exchange (LSE), which has fought off a multitude of bids in recent years, welcomed the Qatari investment.
It said it had a long-standing relationship with Qatari investors based on plans to develop the market in Gulf state.
Qatar Investment Authority
20% of London Stock Exchange
10% of Sweden's OMX
20% of Nasdaq
28% of London Stock Exchange
Controlling stake of OMX
"The exchange believes that, given the strength of Qatar's economy and the development of Doha as a major financial centre, there are significant opportunities to build further this relationship to the mutual benefit of both parties," Europe's largest exchange said.
The LSE sale was part of a flurry of cross-border activity on Thursday that promised to reshape the already shifting landscape of global stock exchanges.
The Qatar Investment Authority also announced on Thursday that it had acquired 9.98 per cent of OMX AB, Sweden's leading stock market.
In a separate, complex agreement, Nasdaq completed a deal with Borse Dubai that would leave the US firm in overall control of OMX and give the Dubai company firm footholds in Nasdaq and the London bourse.
Borse Dubai will complete its $4bn takeover of OMX but will then sell all of its interest to Nasdaq.
In return, Borse Dubai will receive a 19.99 per cent stake in Nasdaq and most of its stake – 28 per cent - in the LSE.
It will also get two of 16 board seats in a combined Nasdaq-OMX.
It is unclear if the Qatar Investment Authority will try to stop Borse Dubai from taking the London stake from Nasdaq and thereby displace Qatar as the largest shareholder.
Political scrutiny could further complicate the issue.
US politicians have raised security concerns over the sale of Nasdaq.
Charles Schumer, chairman of the Joint Economic Committee and a senior member of the Senate Banking Committee, expressed doubts about the deal and asked the treasury department to review the deal.
However, George Bush, the US president, said in Washington that he was concerned protectionism could hamper economic growth.
"We're going to take a good look at it, as to whether or not it has any national security implications involved in the transaction. And I'm comfortable with the process to go forward," he said.
Borse Dubai's voting rights at Nasdaq would be limited to 5 per cent, which may help assuage any concern that a Middle Eastern government-controlled entity would for the first time own a sizable chunk of a US exchange.
An attempt by another state-owned Dubai firm to buy operating rights at six US ports 18 months ago sparked a firestorm of protest in the US congress which ultimately squelched the deal.
Nasdaq said it would submit the current deal to the federal Committee on Foreign Investment which reviews acquisitions of American companies by foreign entities for security concerns.
The transactions are also subject to approval by shareholders in Europe and the US.