They emerged holding hands shortly after the courthouse opened at 8 a.m., as opponents booed and supporters cheered and chanted, "Love wins!"
"We're just really ... happy right now to finally get married and have it recognized here," Yates, who proposed to Smith this summer after a 10-year relationship, said shortly before getting the license.
But County Clerk Kim Davis sent word from the county jail that she considers marriage licenses void unless she approves them, according to her lawyer, Mat Staver.
"They are not worth the paper they are written on," he said at a Friday afternoon press conference after visiting Davis in jail.
Davis had refused to give licenses
to same-sex couples after the Supreme Court decision -- Smith and Yates alone were denied five times, they say -- on grounds that issuing the licenses would violate her Christian convictions against same-sex marriage.
A federal judge ordered her to jail Thursday, ruling she was in contempt of court for refusing to issue the licenses and not allowing her deputies to distribute them for her.
U.S. District Judge David Bunning said Davis would remain behind bars until she complies. Five of her deputies agreed Thursday to issue marriage licenses in her absence, allowing Smith and Yates -- and any other couple -- to pick theirs up Friday.
How long will Davis stay in jail?
Staver said Davis has no plans to resign and would remain in jail until a compromise is reached. He said his client would be willing to issue licenses if her name and title were not on them.
Davis' husband, Joe, told reporters Friday that his wife was willing to stay in jail until that proposed compromise happened.
"As long as it takes," Joe Davis said. "Hopefully (Kentucky Gov. Steve) Beshear will have the guts to do his job."
Staver also criticized Beshear on Friday, saying, "She is incarcerated not because of anything she's done but because of what the governor has failed to do."
He said the governor could issue an executive order to solve the problem.
The state legislature could pass a law removing clerks' names from the licenses, but it won't be in session until January.
Beshear said this week he won't call lawmakers for a special session to deal with the issue, adding that to do so would cost "hundreds of thousands of dollars of taxpayers' money."
Deputies take over
Beshear on Thursday welcomed the news that Davis' deputies agreed to issue the licenses.
"The future of the Rowan County Clerk continues to be a matter between her and the courts. Deputy clerks have said they will commence issuing marriage licenses beginning (Friday)," he said. "It appears that the citizens of Rowan County will now have access to all the services from the clerk's office to which they are entitled."
In court papers, attorneys for Davis argued that she is unable to comply with the court orders because issuing same-sex marriage
licenses "irreparably and irreversibly violates her conscience."
But American Civil Liberties Union attorneys contended Davis has no legal basis to avoid performing her duties as a government clerk.
A federal prosecutor said it was time for Davis and her county to comply.
"Government officials are free to disagree with the law, but not disobey it," U.S. Attorney Kerry B. Harvey said in a statement. "The county clerk has presented her position through the federal court system, all of the way to the U.S. Supreme Court. It is time for the clerk and the ccunty to follow the law."
Bunning said he, too, was religious, but he explained that when he took his oath to become a judge, that oath trumped his personal beliefs, CNN affiliate WKYT-TV reported
"Her good faith belief is simply not a viable defense," Bunning said.
Jailing was 'not what everyone was hoping for'
Staver, founder of Liberty Counsel, which represented Davis, said he was stunned by Thursday's ruling ordering Davis to jail.
"Knowing Kim Davis and her strong Christian resolve and convictions, she may be jailed behind bars, but her conscience remains free," he told CNN's "The Lead with Jake Tapper" on Thursday.
Yates, who received the marriage license with Smith on Friday, said he, too, was shocked by Davis' jailing.
"That's not what everyone was hoping for," Yates said, adding that same-sex marriage supporters were instead hoping Davis would be fined or that she would resign. "It was a shock, but there have been so many things that we didn't anticipate."
Daniel Canon, an attorney who was working with the ACLU on the case against Davis, said his clients had not asked for Davis to be jailed. But now that she is, he said, there should be "some assurance that Ms. Davis is not going to continue to impose her religious beliefs."
Ryan Anderson, a senior research fellow at the Heritage Foundation, a Washington-based conservative think tank, said the state legislature should remove clerks' names from the licenses as Davis has asked.
"Hopefully she'll get out of jail because the state of Kentucky will realize that there are compromises we can reach that will protect both the rights of gays and lesbians to receive marriage licenses and the rights of someone like Kim Davis not to have her name on that marriage license," he said.
Anderson acknowledged that Davis could resign, but he said she shouldn't have to.
"We have a rich history in the United States of accommodating conscientious objectors," Anderson said. "Kentucky accommodates conscientious objectors for other types of licensings. ... The question should be: If we can accommodate someone, why shouldn't we?"
Beshear, the governor, says he has no power to remove Davis from office.
Smith said the license denials had taken an emotional toll. During the fifth and final time he and Yates were rejected, people were protesting against them outside the courthouse, he said.
"We had our hearts broken. That's upsetting enough, and you come out the door and they start cheering and clapping, and I just started crying. I couldn't stop for a while -- I was pretty upset," Smith said.
The couple looked wary as journalists surrounded them as they left the courthouse Friday. One said he didn't anticipate being the first to get the license, but instead believed two other couples had intended to arrive before them.
They said they were happy and elated before asking reporters to make way so they could leave.
A different person
Some scoff at the clerk, suggesting she's a hypocrite because she has been divorced three times
Davis said she's a different person since becoming a Christian 4½ years ago.
"I am not perfect," she said in a statement. "No one is. But I am forgiven."
Staver said Davis occupies a cell by herself. She slept well the first night and has been studying the Bible, he said.
"She has a clean conscience even though she's incarcerated behind these bars," he said.