Federal law generally bars illegal immigrants from being covered by Medicaid. But a little-known part of the state-federal health insurance program for the poor has long paid about $2 billion a year for emergency treatment for a group of patients who, according to hospitals, mostly comprise illegal immigrants. The lion's share goes to reimburse hospitals for delivering babies for women who show up in their emergency rooms, according to interviews with hospital officials and studies. Some groups say the services encourage people to cross the border for care, while advocates for immigrants say the funding is inadequate because it doesn't...