The Democratic-controlled Senate is unlikely to take up the measure, however.
The bill, approved largely along party lines in the House, would prohibit abortions after the 20th week of pregnancy, with limited exceptions to save the life of the mother or in cases of rape or incest, so long as they have been reported to law enforcement.
The final vote was 228 to 196. Six Republicans voted against the measure, while as many Democrats voted for it.
The 1973 Roe vs. Wade decision allowed abortions until a fetus is viable outside the womb, generally considered to be at 24 weeks. The U.S. 9th Circuit Court of Appeals last month struck down an Arizona law that also would have banned abortions after 20 weeks of pregnancy. Other states with Republican-led legislatures have also pushed new abortion restrictions.