The outcome is anyone's guess in a state where abortion has traditionally been a non-issue.
NARAL Pro-Choice America President America Ilyse Hogue said it is the first municipal ballot on abortion that she knows of, and her group is watching the election closely.
"I am concerned every single time these extreme ideologues seek to roll back what we believe is settled law in terms of a woman being able to make decisions about her health," Hogue said.
National organizations on both sides have descended on Albuquerque. Among their key targets are Catholics and Hispanics.
Roman Catholic Archbishop Michael Sheehan of Santa Fe is urging Albuquerque parishioners to vote for the ban on late-term abortions, which does not include any exceptions for things like rape and fetal anomalies. But the national group Catholics for Choice is working to counter that message, both in advertisements and on the ground.
"We have been hearing from lots of Catholics in Albuquerque who are unhappy that this is being pushed through their churches, in what is supposed to be a sacred space," said Sara Hutchinson, domestic program director for the group.
Project Defending Life has close ties to Operation Rescue and its national network. It has brought in volunteers from around the country to contact voters, lead prayers and help with outreach to churches.
Opponents have criticized the tactics used by Project Defending Life.
"Quite frankly I think they are giving misleading and biased information both on the ballot and in the graphic stuff they are using. It is really disrespectful to the voters. ... It doesn't reflect the intelligence of the Albuquerque voters," said Julianna Koob of Planned Parenthood of New Mexico, which is among the groups campaigning against the measure.
Tara Shaver makes no apologies. For example, the moving billboard, she said, shows pictures of "actual abortion victims. Babies that were killed by abortion. We use the images because as everyone says, a picture speaks louder than 1,000 words."
"We want to restore meaning to the word abortion," Shaver said. "And we want people to realize that this election is literally between life and death."
Shaver said she and her husband first came to New Mexico from Wichita, Kansas, in 2011 because it is home to Southwestern Women's Options, one of just a handful of late-term abortion clinics in the country. Initially, she said, they went after the clinic by combing through emergency calls from the clinic for ambulance services and lobbying lawmakers to tighten the state's abortion laws.
If the referendum passes, a legal challenge is expected. Attorney General Gary King, a Democrat, has said he believes the law is unconstitutional.
Asked if other cities with late-term abortion clinics might be targeted in the future, Shaver said, "We are encouraging people to see what can be done at the city level ... We are starting to get calls from people asking us how to do what we have done."