The two are tied at 42% each among registered voters, the Fox News poll
published Tuesday, conducted by Democratic firm Anderson Robbins Research and GOP firm Shaw & Company Research.
Moore and Jones will face off in a special election to fill Attorney General Jeff Sessions' former seat December 12.
The poll confirms Republicans' fears that Moore -- who campaigns on a theocratic, anti-LGBT message
and has twice been ousted as state Supreme Court chief justice, once for refusing to remove a Ten Commandments monument and once for refusing to follow the Supreme Court's ruling legalizing same-sex marriage -- would be a uniquely vulnerable candidate in a state President Donald Trump won last year by 28 percentage points.
Jones, a former prosecutor who convicted two former Klansmen in the bombing of Birmingham's 16th Street Baptist Church, has sought to portray Moore as someone who would embarrass Alabama on the national stage.
"I can work with Republicans better than Roy Moore can work with anyone," Jones says in a new TV ad
The poll found that just 53% of registered voters said they were extremely or very interested in the race. Among those voters, Jones has a one-point lead over Moore at 46% to 45%.
Moore defeated Sen. Luther Strange,
who was appointed to fill Sessions' seat until the special election, in a September 26 primary runoff. His win came despite Trump having visited Huntsville to campaign for Strange. Moore was backed by Steve Bannon, Trump's former White House chief strategist. And he has campaigned on removing Mitch McConnell as Senate majority leader.
Jones, meanwhile, launched his campaign with a Birmingham rally alongside former Vice President Joe Biden.
Moore has a history of under performing compared to national Republican leaders in Alabama. In the 2012 race for state Supreme Court chief justice, he won by just 2 points over little-known circuit court judge Bob Vance in a year Mitt Romney won the state by 22 points.
The survey of 801 voters, conducted October 14-16, has a margin of error of plus or minus 3.5 points.