A majority of Colorado voters support a ballot measure to legalize limited possession of marijuana, according to a new Denver Post poll.

The poll found that the measure, Amendment 64, has the support of 51 percent of likely voters surveyed, compared with 40 percent opposed. Men favor the measure more than women, a common gender split on the issue. But 49 percent of women polled said they support the measure, compared with 39 percent who said they are opposed.

Across every income bracket and in every age group except those 65 and older, more voters told pollsters they support the measure than oppose it, though some of the leads fall within the 4-percentage-point margin of error. Voters younger than 35 support the measure by a margin of 30 percentage points, 61 percent to 31 percent, according to the poll.

The automated telephone poll was conducted Sept. 9-12 for The Post by New Jersey-based SurveyUSA. About 26 percent of those questioned were cellphone-only users, who were shown a questionnaire on their smartphone, tablet or other electronic device. Of voters included in the sample, 34 percent said they are Republicans, 34 percent said they are Democrats, and 30 percent identified as unaffiliated voters.

While several previous polls have found more support for Amendment 64 than opposition, the Post’s survey is the first independent poll to find more than 50 percent support.

That may not hold until Election Day — support for a 2010 marijuana-legalization measure in California polled at 52 percent three months before the election, while the measure ultimately failed with 54 percent opposition. But the results suggest Colorado could be in position to become the first state in the nation to legalize marijuana possession for adults for recreational use.

They also pose a challenge for the re-election campaign of President Barack Obama, who across the country is courting the young voters who support marijuana legalization while remaining opposed to the policy change.

In the run-up to the Democratic National Convention, Obama’s campaign released a video intended to connect with young voters that featured actors John Cho and Kal Penn. The pair are best known for playing the titular stoners of the marijuana-laced “Harold and Kumar” movie franchise. While the video was a hit with some supporters — “Such an awesome video Mr. President,” one commenter wrote on YouTube — it incensed marijuana activists who saw it as hypocrisy.

The conflict is particularly acute in Colorado, not only because of Amendment 64’s presence on the ballot but also because the Justice Department has recently begun taking action to shut down medical-marijuana dispensaries near schools. Banks in Colorado, fearful of federal regulations against working with illegal businesses, have refused to do business with state-legal medical-marijuana businesses.

Banking restrictions forced Wanda James to close her Denver medical-marijuana-products business last month. James, who served on Obama’s national campaign finance committee in 2008 and personally contributed $25,000 along with her husband, said this year she won’t help Obama’s re-election bid.

James said she feels insulted every time a flier asking her to give to Obama’s campaign arrives in her mailbox.

“I was on your finance committee,” James said. “You destroyed my business. Now you want me to give you money?”

James said she will still vote for Obama, hoping that his policies change during a second term. But she notes that, if enough marijuana supporters decide to vote for Libertarian candidate Gary Johnson — who supports marijuana legalization — it could cost Obama the state and the election.

Johnson is, in fact, counting on the pot vote in Colorado. He has held campaign events at medical-marijuana dispensaries in Denver.

“Whether or not that equates to votes, we’ll see,” Johnson said. “But I do view this issue as an example of myself putting issues before politics.”

Johnson — who registers at 3 percent in the Post poll — is right to be cautious about the interplay of marijuana politics and voter turnout. A Post analysis in 2010 found no evidence that marijuana initiatives drive up turnout or influence candidate selection. Daniel Smith, a University of Florida political- science professor, said ballot initiatives can sometimes have an impact on turnout — but almost never in presidential elections, where the top-line race overshadows everything else.

“This marijuana-legalization measure may be bringing some people to the polls,” Smith said, “but very few of them wouldn’t be coming otherwise.”

Colorado marijuana activist Rico Colibri said he expects Amendment 64 to have little impact on the presidential race. Despite disagreeing with Obama’s marijuana policies, Colibri said he plans to cast “a reluctant vote” for him.

“His stance on marijuana certainly isn’t great,” Colibri said. “Then again, he could be taking a much more aggressive stance, and he isn’t.”

How this poll was conducted

This SurveyUSA poll was conducted by telephone in the voice of a professional announcer. The poll has a margin of error of plus or minus 4 percentage points. Respondent households were selected at random, using Random Digit Dialed samples provided by Survey Sampling of Fairfield Conn., unless otherwise indicated on the individual poll report. All respondents heard the questions asked identically. The number of respondents who answered each question and the margin of sampling error for each question are provided. Where necessary, responses were weighted according to age, gender, ethnic origin, geographical area and number of adults and number of voice telephone lines in the household, so that the sample would reflect the actual demographic proportions in the population, using most recent U.S.Census estimates. In theory, with the stated sample size, one can say with 95% certainty that the results would not vary by more than the stated margin of sampling error, in one direction or the other, had the entire universe of respondents been interviewed with complete accuracy. Fieldwork for this survey was done by SurveyUSA of Clifton, NJ.