The current struggle against extreme Islamist terrorism is sometimes characterized as a “clash of civilizations.” More accurately, it is a civil war within Islamic civilization between a radical minority that uses violence to enforce a simplified and ideological version of its religion and a mainstream that has more diverse and tolerant views. Trade, economic growth, education, development of civil-society institutions, and gradual increases in political participation may help strengthen the mainstream over time. Equally important will be the narrative presented by the West. Defeating Islamist terrorism requires hard intelligence and police work, but we must also attract mainstream Muslims to dry up the sources of new radical recruits. Thus far, intelligence reports indicate that the policies of the United States have created more terrorists than it has killed. America needs to combine hard and soft power into “smart” power, as it did during the Cold War.

Many official instruments of soft, or attractive, power—public diplomacy, broadcasting, exchange programs, development assistance, disaster relief, military-to-military contacts—are scattered around the government with no overarching strategy or budget that even tries to integrate them. The United States spends about 500 times more on the military than it does on broadcasting and exchanges combined. The United States cut short-wave, English-language broadcasts to save the equivalent of less than half an hour of the defense budget of the United States. How do we make such trade-offs? And how should the government relate to the nonofficial generators of soft power—everything from Hollywood to Harvard to the Bill and Melinda Gates Foundation—that emanate from our civil society?

The United States should develop a smart-power strategy by creating a deputy national...