National abortion rights groups have sought for years to persuade state legislatures to adopt laws guaranteeing abortion rights as a backup to Roe. But they have had limited success: Only seven states have such measures in place, including California, Connecticut and Maryland; the most recent state to adopt such a law is Hawaii, which did so in 2006.

“Pretty much all of the energy, all of the momentum, has been to restrict abortion, which makes what could potentially happen in New York so interesting,” said Elizabeth Nash, state issues manager at the Guttmacher Institute, a research group that supports abortion rights. “There’s no other state that’s even contemplating this right now.”

In most statehouses, the push by lawmakers has been in the opposite direction. The past two years has seen more provisions adopted at the state level to restrict abortion rights than in any two-year period in decades, according to the Guttmacher Institute; last year, 19 states adopted 43 new provisions restricting abortion access, while not a single significant measure was adopted to expand access to abortion or to comprehensive sex education.

“It’s an extraordinary moment in terms of the degree to which there is government interference in a woman’s ability to make these basic health care decisions,” said Andrea Miller, the president of NARAL Pro-Choice New York. “For New York to be able to send a signal, a hopeful sign, a sense of the turning of the tide, we think is really important.”

Abortion rights advocates say that even though the Roe decision supersedes state law, some doctors are hesitant to perform late-term abortions when a woman’s health is at risk because the criminal statutes remain on the books.

“Doctors and hospitals shouldn’t be reading criminal laws to determine what types of health services they can offer and provide to their patients,” said M. Tracey Brooks, the president of Family Planning Advocates of New York State.

For Mr. Cuomo, the debate over passing a new abortion law presents an opportunity to appeal to women as well as to liberals, who have sought action in Albany without success since Eliot Spitzer made a similar proposal when he was governor. But it also poses a challenge to the coalition of Republicans and a few Democrats that controls the State Senate, the chamber that has in the past stood as the primary obstacle to passing abortion legislation in the capital.