“We have signs that there are positive results,” said Vijay Pillai, the World Bank country manager in Sierra Leone.

In recent years, Zambia, Burundi, Niger, Liberia, Kenya, Senegal, Lesotho, Sudan and Ghana have gone to some form of free care, particularly for pregnant women and young children, Mr. Yates noted two years ago in the health journal The Lancet. Rwanda has been offering nominal rates for health insurance for over a decade, and after fees were dropped in Burundi in 2006, average monthly births in health facilities rose by 61 percent and Caesarean sections went up by 80 percent, he found.

“It’s absolutely common sense that if we increase the consumption of the services,” improvements in health follow, he said. “It’s blindingly obvious. We know these medicines work.”

Still, the hurdles loom large. Here in Sierra Leone, the health minister, Zainab Bangura, says her country needs 54 gynecologists but has only 4. Likewise, she says, there are only two pediatricians in a nation of over five million people. “We lost 10 years” to civil war, Ms. Bangura said of the impetus behind increasing access to health care. “We needed to embark on a drastic measure.”

But donors will not finance the program forever, and the hope is that revenues from the mining of diamonds and minerals, shaky for now, will replace them. Beyond that, Unicef recently discovered that drugs equivalent in value to 14 percent of what it had donated were missing. The agency has demanded an investigation.

Given how recent, untested and strained some of the efforts to provide free health care are, some researchers are reluctant to make an automatic correlation between better access and better health. The outcomes are “not very straightforward,” said Sophie Witter, a senior research fellow at the University of Aberdeen, in Scotland, who has studied the elimination of fees in Africa.