There are some indications that Falivena's baby bonus is succeeding - the first-grade class has 17 students this year - but that figure may be misleading. As it turns out, many of the new parents who have taken advantage of the bonus are locals who planned to have a child anyway. (Ida Robertiello, another of the baby-bonus mothers who sang Falivena's praises for me, admitted that she was already pregnant with her son Matteo when Falivena announced his scheme.) The main effect of the bonus money may be on the timing of births. Last year Falivena was out of office, and the temporary replacement canceled the payments. "I know several women in Laviano who are pregnant now," Daniela told me, and her husband added, with a rakish grin, that couples got busy because they knew Falivena was coming back as mayor, with a promise to restart the payments.

But with close to 50 mothers now eligible, Falivena doesn't know how long he can keep the baby bonus going. And Laviano is still losing population.

DEMOGRAPHICALLY SPEAKING, Laviano is not unique in Italy, or in Europe. In fact, it may be a harbinger. In the 1990s, European demographers began noticing a downward trend in population across the Continent and behind it a sharply falling birthrate. Non-number-crunchers largely ignored the information until a 2002 study by Italian, German and Spanish social scientists focused the data and gave policy makers across the European Union something to ponder. The figure of 2.1 is widely considered to be the "replacement rate" - the average number of births per woman that will maintain a country's current population level. At various times in modern history - during war or famine - birthrates have fallen below the replacement rate, to "low" or "very low" levels. But Hans-Peter Kohler, José Antonio Ortega and Francesco Billari - the authors of the 2002 report - saw something new in the data. For the first time on record, birthrates in southern and Eastern Europe had dropped below 1.3. For the demographers, this number had a special mathematical portent. At that rate, a country's population would be cut in half in 45 years, creating a falling-off-a-cliff effect from which it would be nearly impossible to recover. Kohler and his colleagues invented an ominous new term for the phenomenon: "lowest-low fertility."

To the uninitiated, "lowest low" seems a strange thing to worry about. A few decades ago we were getting "the population explosion" drilled into us. The invader species homo sapiens, we learned, was eating through the planet's resources and irretrievably fouling and wrecking its fragile systems. Has the situation changed for the better since Paul Ehrlich set off the alarm in 1968 with his best seller "The Population Bomb"? Do current headlines - global food shortages, climate change - not indicate continuing signs of calamity?

They do, as far as some are concerned, but things have changed somewhat. For one thing, around the world, even in developing countries, birthrates have plummeted - from 6.0 globally in 1972 to 2.9 today - as populations have shifted from rural areas to cities and people have adopted urban lifestyles, and the drop has perhaps lessened the urgency of the overpopulation cry. Meanwhile, in recent years another chorus of voices has sounded. Yes, we're straining resources, they say, and it's undeniable that some parts of the globe are overrun with humanity. But other regions now confront a very different fate. In Europe, "lowest low" isn't just a phenomenon of rural areas like Laviano. Cities like Milan and Bologna have recorded some of the lowest birthrates anywhere, in part because the high cost of living forces couples either to move or to have fewer children. After the term was invented, "lowest-low fertility" got the attention of leaders in Brussels and national capitals across the Continent - and by now everyone from Seville to Helsinki seems to be aware of it. In Greece, the problem is so well situated in the national psyche that it is conversationally compacted: people refer simply to "the demographic." Putting the numbers in a broader world-historical context stirred a debate about Europe's future. Around the time that President Kennedy went to Germany and gave his "Ich bin ein Berliner" speech, Europe represented 12.5 percent of the world's population. Today it is 7.2 percent, and if current trends continue, by 2050 only 5 percent of the world will be European.

To many, "lowest low" is hard evidence of imminent disaster of unprecedented proportions. "The ability to plan the decision to have a child is of course a big success for society, and for women in particular," Letizia Mencarini, a professor of demography at the University of Turin, told me. "But if you would read the documents of demographers 20 years ago, you would see that nobody foresaw that the fertility rate would go so low. In the 1960s, the overall fertility rate in Italy was around two children per couple. Now it is about 1.3, and for some towns in Italy it is less than 1. This is considered pathological."