Correction Appended

AUSTIN, Tex. — Unable to afford health insurance, Dee Dee Dodd had for years been mixing occasional doctor visits with clumsy efforts to self-manage her insulin-dependent diabetes, getting sicker all the while.

In one 18-month period, Ms. Dodd, 38, was rushed almost monthly to the emergency room, spent weeks in the intensive care unit and accumulated more than $191,000 in unpaid bills.

That is when nurses at the Seton Family of Hospitals tagged her as a “frequent flier,” a repeat visitor whose ailments — and expenses — might be curbed with more regular care. The hospital began offering her free primary care through its charity program.

With the number of uninsured people in the United States reaching a record 46.6 million last year, up by 7 million from 2000, Seton is one of a small number of hospital systems around the country to have done the math and acted on it. Officials decided that for many patients with chronic diseases, it would be cheaper to provide free preventive care than to absorb the high cost of repeated emergencies.