YAUTEPEC, Mexico  When her twin girls were born seven weeks early, Azucena Mora Díaz did not have to worry about how she would pay for expensive hospital care, even though her husband has only a low-wage job as a construction worker’s assistant.

Under a government insurance plan for the poor, the girls were treated at the Women’s Hospital here and continue to receive follow-up care to monitor their development. The couple pays nothing.

“We owe everything to this,” Ms. Mora said as one of the twins, now 13 months old, squirmed in her arms, wearing a T-shirt emblazoned with a single word: “Smile.”

A decade ago, half of all Mexicans had no health insurance at all. Then the country’s Congress passed a bill to ensure health care for every Mexican without access to it. The goal was explicit: universal coverage.