The prices have been record breaking, and startling.

More than $70 million for Rothko’s “White Center” in 2007, a high-water mark for that artist.

More than $20 million later that year for a Damien Hirst pill cabinet, then a record for a living artist.

And $250 million for Cézanne’s “Card Players” in 2011, the highest known price ever paid for a painting.

Given the secrecy of the art market, few knew at the time who had laid out such unprecedented sums.

But it has become increasingly clear that those masterpieces and many more have been purchased by Qatar, a tiny Persian Gulf country with enormous wealth and cultural ambitions to match: it is buying art at a level never seen before.

“They’re the most important buyers of art in the market today,” said Patricia G. Hambrecht, the chief business development officer for Phillips auction house. “The amount of money being spent is mind-boggling.”

The purchasing is directed through intermediaries by Sheika al Mayassa bint Hamad bin Khalifa al-Thani, chairwoman of the Qatar Museums Authority and a sister to Qatar’s new emir. At age 30 she has become one of the most influential players in the art world.

No one knows exactly how much Sheika al Mayassa has spent on behalf of her family or the museum authority since she was named chairwoman by her father, the former emir, in 2006. But experts estimate the acquisition budget reaches $1 billion a year and say the Qataris have used it to secure a host of undisputed modern and contemporary masterpieces by Francis Bacon, Roy Lichtenstein, Andy Warhol and Jeff Koons.

Where all this art will eventually end up remains something of a mystery. But it seems clear that, just as Qatar has used its oil riches to boost its influence in the Middle East with ventures like arming Syrian rebels, its wealth is also being deployed to help the country become a force in the world of culture.

This effort to create a first-class contemporary art collection, essentially from scratch, has buoyed the international art market, experts say, and contributed to some of the escalation in prices.

Until Qatar’s 2007 purchase, for example, the most expensive Rothko ever sold at auction (“Homage to Matisse”) had drawn $22 million in 2005, less than one-third of the price Qatar paid. In 2011 the $250 million spent for “Card Players” was four times the highest public price ever paid for a work by that artist.

“When they finish their buying program and withdraw from the market,” said David Nash, a New York dealer who spent 35 years as a top executive with Sotheby’s, “they will leave a big hole which I don’t see anyone else ready to fill at their level.”

In recent years the Qatar Museums Authority has created three high-profile museums in the capital, Doha, by the architects Jean Nouvel, I. M. Pei and Jean-François Bodin. But each of these projects — a new home for the National Museum of Qatar now under construction; the Museum of Islamic Art; and Mathaf: Arab Museum of Modern Art — is focused on regional art and artists. So experts expect that a good portion of the Western collection being amassed will become part of a new contemporary art institution in the country, though officials have yet to announce that.

The annual acquisition budgets of major museums typically amount to just a small fraction of what Qatar is spending. The Museum of Modern Art, for example, spent $32 million to acquire art for the fiscal year that ended in June 2012; the Metropolitan Museum of Art, $39 million.

While other gulf states like Abu Dhabi and Dubai are also trying to become cultural capitals, those two members of the United Arab Emirates have teamed up with existing institutions — namely the Louvre and the Guggenheim — to establish themselves. Qatar, meanwhile, is going it alone.

“They see themselves as an international center for many cultures,” said Allen L. Keiswetter, a scholar at the Middle East Institute in Washington. “It establishes them as another reason to be a destination for travel, for business. If you want to attract people, you need to have a reason to go there.”

Sheika al Mayassa declined to be interviewed for this article, but she has made limited remarks about the role art will play in Qatar’s future.

“We are revising ourselves through our cultural institutions and cultural development,” she said in a 2010 TED Talk. “Art becomes a very important part of our national identity.”

In an interview that year with The New York Times, the sheika suggested that establishing art institutions might challenge Western preconceptions about Muslim societies.

“My father often says, in order to have peace, we need to first respect each other’s cultures,” she said. “And people in the West don’t understand the Middle East. They come with Bin Laden in their heads.”

Sheika al Mayassa appears to combine Western and Muslim influences. Sometimes she dresses like a stylish business executive; sometimes she wears a traditional black abaya, which covers everything but her round face. She speaks French and English as well as Arabic.

The sheika does not have a formal background in art history, having studied political science and literature at Duke University in North Carolina. Both she and her husband, Sheik Jassim bin Abdulaziz al-Thani, also did postgraduate work at Columbia University before returning to Qatar.

“The sheika has a very grand vision and is a very educated woman,” said Leila Heller, a New York dealer with many Middle Eastern artists. “She wants to make Doha a hub for art in the region, where people don’t necessarily have to fly to New York and to Paris and to L.A. to see great shows. Doha has an ambitious plan of opening close to 20 museums of different kinds.”

As to the guiding principle behind the buying, art experts say the sheika is simply trying to amass the best of the best, whatever the price. The pace of her purchasing during the past three or four years has vaulted her, many say, to a place among the art world’s most powerful figures. She has been written about by publications like Forbes and Vanity Fair; The Economist recently pronounced her “the art world’s most powerful woman.”

And yet her profile within the places where art is bought and sold is all but nonexistent. The sheika does not visit art galleries. She won’t be found in the auction room on sale nights among the other major collectors for whom the business is something of a high-society fraternity.

Auction houses and galleries will often wine and dine these collectors before a big sale. Significant works will be flown to their homes. The Qataris don’t take part in this. They delegate their purchasing to a handful of experienced art advisers who do it for them — initially the dealers Philippe Ségalot and Franck Giraud and now Guy Bennett.

Mr. Bennett, a former co-head of Christie’s Impressionist and Modern art department worldwide, is known as a master dealmaker. He is just one of a number of Christie’s alumni who have taken on roles with Qatar. In June 2011 Edward Dolman, the auction house’s former chairman, was named executive director of the Qatar Museums Authority. Jean-Paul Engelen, the director of public art programs for the authority, is also a Christie’s veteran.

The Qatar team typically buys from dealers, though some of its most major purchases have been at auction. Experts said that another intermediary may act on Mr. Bennett’s behalf so that the sales cannot be traced to the Qataris, who want to keep their buying private to prevent driving up the market and fueling speculation about their plans.

“They are very secretive about their purchases and activities in the art market and I am not quite sure why,” Mr. Nash said.

American art institutions could be expected to be frustrated to see so many important pieces leaving the United States. But some museum executives say that’s just how it goes.

“Sure, there are lots of works of art that we have absolutely wanted,” said Glenn D. Lowry, director of the Museum of Modern Art. “But if Rothko and de Kooning and Kline, among others, end up circulating in Moscow, Qatar and Shanghai, that’s not so bad — it’s a projection of American culture and importance that is meaningful.”