Utah is the first state to take this action. The justices will decide whether to hear the arguments later this year and if they take the case, they'll likely rule in mid-2015.
The poll, conducted last week by Dan Jones and Associates, found that 63 percent of respondents supported the state's decision to seek a Supreme Court ruling, while 58 percent believe the high court will side with the district court and appeals-court judges and legalize gay marriages.
The poll also found that 61 percent of Utahns oppose gay marriage, while 29 percent support it. The survey involved 400 likely voters and had a margin of error of 4.9 percent
In 2004, 66 percent of voters approved Amendment 3.
That's not much of a decline in the past 10 years, according to Derek Monson, the director of public policy at the conservative Sutherland Institute.
"Despite all of the full-court press on the other side to say everybody who believes in traditional marriage is behind the times the reality is the opposite," he said. "Here in Utah we believe in traditional marriage and we support elected officials who believe in traditional marriages."
Monson said he believes Utah has a chance to prevail at the Supreme Court.
"Nobody knows what they are actually going to do, but we think we have a good shot, we think we have a fair chance," he said.
Utah's population is about 60 percent Mormon, and the LDS faith is opposed to same-sex marriage. The UtahPolicy poll found that 88 percent of respondents who identified themselves as "very active" Mormons opposed gay marriage.
"I think it's fair to say this is an incredibly delicate issue that is close to people's hearts," said Marina Gomberg, the interim executive director of Equality Utah, a gay rights organization. "We have to understand that the change in attitudes sometimes happens slower than the changing of laws."
Gomberg sees the five percentage point decline between the 2004 vote and the UtahPolicy poll as a step toward more acceptance of gay marriage.
"We are moving in a positive direction, one that favors equality and fairness and opportunity," she said.
The findings of the UtahPolicy poll are quite different than one sponsored by The Salt Lake Tribune in January, just weeks after U.S. District Judge Robert Shelby struck down Utah's gay marriage ban. The Tribune poll, conducted by SurveyUSA, posed the question in a slightly different way. Instead of asking if Utahns supported gay-marriage, the Tribune poll asked whether same-sex couples in Utah should be allowed to get state-issued marriage licenses. Respondents were split evenly, with 48 percent saying yes and 48 percent saying no.
When asked if these couples deserved any form of civil unions or domestic partnerships, support rose to 72 percent, and that included 65 percent of Mormon respondents.