Franks's bill would ban abortions after 20 weeks' gestation based on the premise — which is disputed — that fetuses can feel pain at that stage. It would exempt women whose lives are in danger, but not victims of rape or incest.
A D.C.-only version failed to pass the House last year under suspension of the rules, which requires support from a two-thirds majority of lawmakers.
Abortion-rights activists vowed to deliver a similar defeat this year.
"It is astonishing that the Subcommittee on the Constitution would support such a clear affront to the U.S. Constitution — especially when everywhere similar laws have been challenged in the courts, they have been immediately blocked," said Nancy Northup, president and CEO of the Center for Reproductive Rights, in a statement.