The measure would prohibit abortions in the District of Columbia after 20 weeks' gestation based on the disputed premise that fetuses can feel pain at that stage.
Last year's bill exempted women whose lives are in danger but not victims of rape or incest.
"The gruesome late-term abortion of unborn children who can feel pain is the greatest atrocity in the United States today," Franks said in July.
He added that science "incontrovertibly demonstrates" that fetuses can feel pain before viability.
Republicans have tried a handful of times over the last two years to restrict abortion rights in the District.
Those measures, besides sparking conflict between established abortion foes, picked up additional resistance from supporters of D.C. home rule.
That movement says Congress should maintain precedent and not interfere with the District on matters of local policy.
Congressional Republicans say the constitution allows for federal lawmakers to govern the District as they see fit.
Franks's bill, as introduced last time, would have imposed fines and prison terms of up to two years for D.C. doctors who perform abortions after 20 weeks.
It also created new reporting requirements and authorized partners and parents of women who have abortions to petition for civil remedies.
Opponents called the measure "flagrantly unconstitutional" because it targeted D.C. and contradicted Roe v. Wade by prohibiting abortions before fetal viability.
"This is nothing more than a direct challenge to Roe v. Wade and a vehicle for yet another ideological attack against women's reproductive rights," Rep. Barbara Lee (D-Calif.) said on the House floor last summer.