The campaign against birthright citizenship has been on a roll. Last month, it won the endorsement of South Carolina Republican Sen. Lindsey Graham, long seen as a moderate on immigration. He favors amending the Constitution because he's aghast that "people come here to have babies."

Graham and his allies got a boost last week from the Pew Hispanic Center, which released a report that sparked ominous headlines. "Illegal immigrants bear 8 percent of children born in the U.S.," blared Fox News. "Rise seen in births to illegal dwellers," proclaimed USA Today.

Sweet vindication, right? Not quite. In fact, the Pew study refutes the case being made against granting citizenship to children born here to illegal immigrants. It shows that the anti-immigration crowd is chasing a chimera.

Pew did estimate that of the 4.3 million babies born here in 2008, 340,000 had at least one parent who was an illegal immigrant. It also found that "nearly half of unauthorized-immigrant households are couples with children."

But how many "come here to have babies"? Not many. Jeffrey Passel, who co-authored the report, told me that "85 to 88 percent of the mothers have been in the U.S. for at least a year," and "a majority have been here at least three years."

Someone who has a child a year or three after arriving is not exactly in line with Graham's image of pregnant Mexicans wading the Rio Grande in search of the nearest maternity ward. At most, only 15 percent of the mothers arrived here in a mode of expectancy.

But even that modest figure overstates the alleged problem. The kids referred to in the study are those with at least one illegal parent, and many of those parents are married to legal residents. If one parent is a U.S. citizen and you're born on U.S. soil, you'd be a citizen even if Graham got his way. Only a portion of that 12 to 15 percent would be barred.

The more sober opponents of illegal immigration don't think birthright citizenship is much of a draw. When I called Ira Mehlman of the Federation for American Immigration Reform, he said, "It's probably one factor among many. Most people come because of better economic opportunities." Change the Constitution, and they'd keep coming.

Why are all these undocumented foreigners producing offspring on U.S. soil, if not because of birthright citizenship? Some obvious explanations: Because they live here, and because they tend to be of childbearing age, since older folks are less likely to trek through the desert for the privilege of harvesting watermelons.

But the chief reason is that having kids is what human beings do, wherever they are and whatever their immigration status. The odd thing would be if these newcomers weren't reproducing.

You don't need an incentive to get them to bear children, any more than you need artificial inducements to get college students to drink beer. Changing the citizenship rule would have little or no effect on the fertility of illegal immigrants.

There is one category of foreigners who do travel to this country just to give their babies the gift of American citizenship. "Birth tourists" reportedly are catered to by travel agencies, hospitals, and hotels offering deals for expectant mothers. One Turkish-owned hotel in Manhattan offers a package including month-long accommodations for $45,000, which doesn't cover hospital costs.

But it's hard to see why this phenomenon causes so much anger among anti-immigrant groups. They complain about poor Mexicans sneaking in illegally, taking jobs, getting government assistance, and staying forever. Shouldn't they be happy to hear about rich Turks getting visas, avoiding employment, paying their own costs, and leaving immediately?

At any rate, it's not exactly a raging epidemic. The National Center for Health statistics says only 7,670 babies were born in 2006 to women who said they don't live here, or a microscopic 0.17 percent of all live births.

If we want to head them off, we don't have to amend the Constitution. We can just deny tourist visas to visibly pregnant women, or make it a federal offense to come here solely to have a baby.

But what's the fun in addressing a modest problem with a minor change? Anti-immigration zealots would much rather mount a heroic expedition to conquer a mighty mountain. Even if it's really a molehill.

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