Hidden in the fine print of the new federal budget is the death of a federal agency. Maybe.

The agency is called the United States Travel Service. It spends a tiny fraction of the budget, $8 million, employes 75 people and takes up one out-of-the-way hallway in the Department of Commerce building. President Carter wants to do away with it.

But nobody should start mourning yet. Carter has been trying to kill the travel service for three years. President Ford tried, too. So far, the agency is still alive.

The reason it has survived is right out of the civics books. An "iron triangle" made up of the agency, the private industry it serves and the congressional committee that oversees it has stymied administration attempts. For every federal program, such an iron triangle exists, and that is why budget-cutting -- as promised by Ronald Reagan and his new budget director, David A. Stockman -- almost never works.

In itself, the United States Travel Service is not terribly important. But its recent history is a classic case of the politics of budget-cutting and, as such, presages the battles the new adminstration will fight if it does, in fact, move to chop federal spending.

These battles are guaranteed to share one theme: nobody wants to cut any program as badly as the people who want to preserve it. "Any fact you see in the budget," said one former OMB official who unsuccessfully battled the travel service, "is somebody else's meat."

Congress created the service in 1961 to encourage foreigners to take vacations in the United States through advertising and public relations activities. Twenty years later the debate about the agency is remarkably straightforward. Everybody agrees it is good for America when foreigners spend money here. But they disagree whether promoting that should be the job of government -- as it is in most western nations -- or of airlines, resorts, travel agencies and other components of the private travel industry.

As the years passed, the U.S. Travel Service made some friends. In 1969, for example, two trade associations merged to form the Discover America Travel Organizations, last year renamed Travel Industry Association of America. This organization became a staunch supporter of the travel service.

In 1971 Congress passed amendments to the International Tourism Act that elevated the person in charge of the U.S. Travel Service to the status of assistant secretary of Commerce. That guaranteed the agency a second powerful defender.

In 1972 the Senate Commerce Committee established a subcommittee on foreign commerce and tourism to oversee the service, among other things. Its first chairman was Sen. Daniel Inouye, a Democrat from tourism-rich Hawaii and, in the words of the travel industry's chief lobbyist, "a great friend of tourism." The iron struggle was complete.

Like the industry group and the assistant secretary, Inouye believes there should be a U.S. Travel Service. That position also happens to be in his self-interest, as it is in the industry's and the assistant secretary's.

By 1975, Ford's Office of Management and Budget tried to cut the service's budget. That did not work. By 1979 the agency had a comfortable budget of $13.5 million and a staff of 20.

Then disaster struck. carter's OMB proposed eliminating the agency's six overseas offices, cutting its Washington staff and folding what was left into another division of the Commerce Department.

The triangle swung into action. Inouye held hearings. The industry lobbied. The travel service made its view known. After all that, Congress passed a bill keeping the overseas offices, keeping the travel service in existence and cutting $5 million and 60 jobs from its Washington office. Carter signed the bill.

The next year, 1980, OMB tried again to fold the travel service into another agnecy and cut its budget. Inouye held hearings again. Congress passed another bill, Carter signed it and the service remained alive and uncut. This year's Carter budget -- in this respect a triumph of hope over experience -- again eliminates the travel service and cuts its spending from $8 million to $3 million.

The friends of the travel service have not, through all these struggles, adhered to a purely defensive posture. In 1978 Inouye introduced legislation authorizing a National Tourism Policy Study. Two consulting firm reports and two sets of hearings later, in 1979, the Senate passed the National Tourism Policy Act, introduced by Inouye. The House passed its version in 1980, and when the smoke cleared a bill was out of Congress making the travel service an independent agency -- free of the Commerce Department, free of prior OMB review of its budget and presumably free to increase its size and spending.

Carter pocket-vetoed the bill last month. In the meantime, when the Democrats lost the Senate, Inouye lost his subcommittee. But the new chairman, Sen. Larry Pressler (R-S.D.), says he will reintroduce the National Tourism Policy Act on Monday, and chances are it will pass.

So Reagan and Stockman will most likely be confronted with two clear options: eliminate the U.S. Travel Service, as the Carter budget recommends, or expand it, as Congress wants. Reagan has not spoken out on the subject; budget office head Stockman, as a congressman, consistently voted with the agency-killers.

Stockman, then, could be a problem, but much of the work of Washington is solving such problems. "One would expect he'd continue" to oppose the travel service, said William D. Toohey, president of the Travel Industry Association, "unless he has the opportunity for" -- he pauses, searching for the right word -- "enlightment."