The health care law has opened up an unusual opportunity for some mothers-to-be to save on medical bills for childbirth. Lower-income women who signed up for a private policy in the new insurance exchanges will have access to additional coverage from their state’s Medicaid program if they get pregnant. Some women could save hundreds of dollars on their share of hospital and doctor bills. Medicaid already pays for nearly half of U.S. births, but this would create a way for the safety-net program to supplement private insurance for many expectant mothers. Officials and advocates say the enhanced coverage will be...