His Highness Prince Karim, the fourth Aga Khan and 49th hereditary imam of the world’s 15 million Shia Imami Ismaili Muslims, remains a paradox to many people. The Pope of his flock, he also possesses fabled wealth and inhabits a world of marvelous châteaux, yachts, jets, and Thoroughbred horses. To be sure, few persons bridge so many divides—between the spiritual and the material; East and West; Muslim and Christian—as gracefully as he does.

Born in Geneva, brought up in Nairobi, educated at Le Rosey and Harvard, the Aga Khan has a British passport and spends a great deal of his time aloft in his private aircraft, but his base is Aiglemont, a vast estate near Chantilly, 25 miles north of Paris. On-site, in addition to a château and an elaborate training center for about a hundred of his Thoroughbreds, is the Secretariat, a modern office block that houses the nerve center of what might be described as his own U.N., the Aga Khan Development Network. A staggeringly large and effective organization, it employs 80,000 people in 30 countries. Although it is generally known for the nonprofit work it does in poor and war-torn parts of the globe, the A.K.D.N. also includes an enormous portfolio of for-profit businesses in sectors ranging from energy and aviation to pharmaceuticals, telecommunications, and luxury hotels. In 2010 these generated $2.3 billion in revenue. The extent of these endeavors might not be so well known to the general public, since the Aga Khan usually shuns the press and stays out of the public eye.

Though he has no political territory, the Aga Khan is virtually a one-man state and is often received like a head of state when he travels. As imam he is responsible for looking after the material as well as spiritual needs of his followers, who are scattered in more than 25 countries across Asia, Africa, the Middle East, Europe, and North America. His projects, however, benefit people of all faiths.

One of the rare opportunities to catch a glimpse of him occurs on a certain Sunday in June, in Chantilly, at the annual Prix de Diane, which for more than a century has been the most prestigious horse race in France. It takes place pretty much in his backyard, at the historic Hippodrome de Chantilly, just a few kilometers from Aiglemont. Dating from 1843, the Prix de Diane is the high point of the Continental horse-racing calendar, on the turf and off. Members of France’s top horse-owner clans, such as the Wildensteins and the Wertheimers, typically appear, along with sheikhs from Qatar and Dubai, and glamorous women in heavily feathered headgear.

Had it not been for the Aga Khan, however, this storied racetrack would probably not exist today, and its surroundings might be heading to ruin. In a highly unusual arrangement, the Aga Khan adopted, you might say, the entire 20,000-acre Domaine de Chantilly, which also contains one of France’s foremost but relatively unknown cultural treasures, the Château de Chantilly. Somewhat ironically, he is using expertise gained in his development projects from Kabul to—literally—Timbuktu to rescue this lush swath of France.

‘His Highness will see you now,” an assistant informs me in the cool white marble lobby of the Secretariat, then ushers me down a long corridor and through what appears to be a heavily fortified door. (Though his closest friends call him “K,” the Aga Khan, 76, is referred to by most of his associates as “His Highness,” “H.H.” for short.)

The Aga Khan’s private office is a large room of minimalist-modern design, with one unexpected feature. Colorful, highly polished spheres—geological specimens from around the world—appear to be floating on the walls, wizard-like.