FORTUNE — As Uncle Sam sees it, even a $135-million museum building designed by one of the world’s greatest architects is in need of a shakedown cruise.

For 39 years, an obscure federal program run by the National Endowment for the Arts has insured — free of charge — billions worth of precious artwork shown in traveling exhibitions around the country. By assuming this risk, the program has saved the hosting museums an estimated $375 million in insurance costs, helping make such shows possible.

But there’s one major caveat to the program: The feds won’t insure the first exhibit in a new or renovated building, no matter its architectural pedigree.

So it was in Fort Worth, Texas, when the long-awaited addition to the Kimbell Art Museum opened late last year. The generously endowed Kimbell owns an exquisite collection of masterworks and occupies a revered building designed by the late Louis Kahn, featuring barrel-vaulted art galleries. Since its opening in 1972, the Kimbell has often been described as America’s finest small museum.

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The problem is that it was too small. The arrival of visiting exhibits regularly required the Kimbell staff to stash much of its own collection in storage, out of view for weeks at a time.

After two decades of debate — driven by fears of desecrating Kahn’s jewel — a resolution to this problem was finally reached when the museum commissioned a separate 101,300-square-foot structure, located across a lawn from the Kahn building and designed by Renzo Piano, the famed Italian architect behind more than 20 high-profile museum projects. The Piano Pavilion’s galleries were specifically built to provide space for traveling exhibits, allowing the Kimbell’s permanent collection to remain in its original home.

But when the Piano building opened in November, it featured much of the Kimbell’s relocated permanent collection, while a recently arrived visiting exhibit from the Art of Institute of Chicago, “The Age of Matisse and Picasso,” occupied its usual space in the Kahn building.

This arrangement — call it the Multi-Million-Dollar Artwork Shuffle — resulted from a quirk of the NEA’s indemnity program, which won’t (as program guidelines state) insure “inaugural exhibitions in new buildings or substantially renovated spaces.”

To deal with this prohibition — while still taking advantage of the government insurance program — Kimbell director Eric M. Lee hung the Chicago art (including 10 Matisses and 10 Picassos) in the Kahn building, while moving his own precious works (including a Michelangelo, a Rembrandt, a Goya, and a Caravaggio) over to the new Piano structure.

The savings was considerable. “Insurance is a huge figure,” notes Lee, who declined to specify the amount.

Temporarily relocating 71 paintings from the Kimbell’s permanent works took about two weeks, as staff moved the paintings slowly on carts through a quarter-mile, climate-controlled tunnel connecting the two structures. When the Chicago exhibit departs in mid-February, the permanent collection will return to the Kahn building, presumably for good. Future visiting exhibits will then appear in the Piano Pavilion, as originally planned — with the benefit of government insurance.

Patricia Loiko, NEA’s indemnity program administrator, says the new-building restriction is imposed “in an abundance of caution.” She offers a list of the potential hazards that delicate artworks face in a new structure: climate-control issues, security, protection from excess light sources, fire-suppression systems, and “off-gassing” of chemicals from new carpet or wood finishes.

“We need to make sure the material is flushed out and doesn’t cause any damage to the artwork,” says Loiko. “We just want to make sure there’s no undue risk to the artwork or to the U.S. Treasury.”

While such risks at a museum like the Kimbell may seem small, the financial stakes are huge. With relatively modest deductibles capped at $500,000 for even a total loss, the government insures approved individual international exhibits up to a value of $1.2 billion; exhibits from U.S. museums are insured for up to $750 million.

Unlike, say, State Farm, the NEA program has no capital of its own to pay out claims. While any loss is backed by the “full faith and credit of the United States,” according to program documents, if any indemnified objects are damaged or lost, the NEA must seek a special appropriation from Congress to make the art-lender whole.

Getting Congress to agree on writing a check for anything might seem a shaky proposition these days. Fortunately, that hasn’t really been tested. In the 39-year history of the program, only two claims have ever been paid — and neither can be blamed for ballooning the federal budget deficit.

The first occurred in 1982, when two paintings by Reuven Rubin, a Romanian-born painter, were reported stolen while being shipped back to Israel from an exhibit at the Jewish Museum in New York City. After a two-year investigation failed to locate the art, the government paid the owners $100,000 to settle the loss. The paintings were later recovered, and the entire $100,000 (after much legal wrangling) was eventually repaid to the U.S. Treasury.

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The second claim resulted from $29,700 in damage in 1988 to two sculptures by Jean Arp, a European artist, while they were on loan to the Minneapolis Institute of Arts. After the deductible, the bill to the U.S. Treasury was a mere $4,700.

All in all, it’s a track record any insurance company CEO would die for.

More than 1,200 exhibits, displaying 120,000 pieces of art, have been insured under the NEA program, says Loiko. “It’s a remarkable loss ratio.”

While Kimbell director Lee believes that “there are lots of issues that a building has that could potentially damage works of art,” he feels confident that his new building, in reality, poses little risk. “Renzo Piano has built so many museums, you pretty much know you’re not going to have any problems,” notes Lee.

Having said that, he’s not about to complain about the unusual provision in the NEA program and says relocating his permanent collection temporarily to qualify the visiting exhibit for government insurance was a no-brainer.

“It makes very expensive exhibitions more affordable,” he notes. “It saved us a lot of money.”