Borse Dubai agreed today to take a stake in the Nasdaq stock exchange and become the first government-controlled stock exchange of the Middle East to hold a significant stake in an American rival, a step that may stir political controversy in Washington.

In a complex four-way deal, Borse Dubai plans to take a 19.9 percent stake in Nasdaq and buy Nasdaq’s 28 percent stake in the London Stock Exchange. Dubai will in turn let Nasdaq buy OMX, the Stockholm-based stock exchange operator, over which the two have been locked in a six-week bidding war. In a separate statement, Qatar, which competes with Dubai as a financial center in the region, said it bought 20 percent of the London exchange but does not currently intend to make an offer.

The agreement allows Borse Dubai, the fastest-growing financial center in the Persian Gulf, to gain access to expertise for its one-month-old enterprise and help it develop into a capital market center in the developing world. For Nasdaq’s chief executive, Robert Greifeld, the deal means he can finally fulfill his goal of creating a more global stock exchange after he failed to combine with the London exchange in the past while the rival NYSE Group successfully merged with Euronext.

“Our primary objective is to build a world class, growth oriented exchange out of Dubai and to become the center for capital markets activities in the emerging markets,” Borse Dubai’s chairman, Essa Kazim, said in a statement today.