Fifty percent of respondents said they support Jones, a former federal prosecutor, while 47 percent said they support Moore, a former Alabama chief judge. That is well within the poll's 4.5-point margin of error, but the survey does show that the allegations of sexual misconduct have severely hurt the Republican candidate who was once seen as the favorite to easily win the race.
President Donald Trump has also said the decision should be up to Alabama voters, discounting the allegations against Moore and saying that electing a Democrat to the seat formerly held by Attorney General Jeff Sessions "would be a disaster".
About 40 percent of women from either party have doubts about the allegations, with 18 percent saying they are "definitely false" and 22 percent saying they're "probably false".
Moore's opponent is Democrat Doug Jones.
The Senate will deal with Moore if he wins, McConnell told ABC's This Week on Sunday. Moore leads among men by 15%.
Three late-November polls, however, show Moore leading Jones. So, when a candidate vehemently denies all of the allegations of inappropriate sexual conduct, including misconduct with a 14-year-old, and is supported by a wife who agrees that all of her husband's accusers are liars, that candidate may have a good chance of getting elected.
Roy Moore and Doug Jones deadlocked in Alabama Senate race, new poll shows
The Washington Post published a bombshell report last month on the allegations based on interviews with more than 30 people.
At the time, he joined a growing number of lawmakers who were calling for Moore to step aside in the race.
Thirty-five percent of respondents to the Post's poll said they think Moore did make unwanted advances toward teenage girls, while 37 percent said they were unsure or had no opinion.
Nine women reportedly have made allegations against Moore.
The CBS/YouGov poll found that 36 percent of the state's Republican voters said the allegations are "definitely false" and another 35 percent said they're "probably false".More news: Mueller removed top Federal Bureau of Investigation agent over possible anti-Trump texts
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