The order came just hours after the Kentucky's attorney general asked for a 90-day delay. The two-page filing says the delay is sought to give that office time to decide whether to appeal the Feb. 12 ruling and would give the state an opportunity to prepare to implement the order.
Heyburn's final order did not mention the request for a stay and he had not ruled on it as of mid-afternoon Thursday.
Earlier this month, Heyburn concluded that the ban, which has been in place since 2004, treated "gay and lesbian persons differently in a way that demeans them."
Dawn Elliott, an attorney for one of the couples pursuing recognition of a marriage performed in Canada, praised the ruling.
"It's a great day to be from the Commonwealth of Kentucky," Elliott said. "I hope that the attorney general and governor that I voted for, don't jump on the appeal bandwagon."
The order means same-sex couples are allowed change their names on official identifications and documents and obtain any other benefits of married couples in Kentucky. Heyburn's ruling doesn't affect a related lawsuit seeking to force the state to issue marriage licenses to same-sex couples. Final briefings in the marriage license case are due to Heyburn by May 28.
It was unclear if or how many people would seek to immediately take advantage of the rights recognized in the rulings. Elliott and co-counsel Shannon Fauver said their clients were considering doing so Thursday afternoon, but had not decided.
Nore Ghibaudy, a spokesman for the Jefferson County Clerk of Court, said until the state issues a directive notifying clerks of the legal change, no same-sex name changes or other legal documents will be issued.
"We have to follow the law until we hear otherwise," Ghibaudy said. "Whatever it is, we'd have no problem doing it."
Allison Martin, a spokeswoman for Kentucky Attorney General Jack Conway, said the office was reviewing the ruling and would have a statement later Thursday.
The decision in the socially conservative state comes against the backdrop of similar rulings or actions in other states where same-sex couples have long fought for the right to marry. Kentucky's constitutional ban was approved by voters in 2004 and included the out-of-state clause.
On Wednesday, a federal judge in Texas struck down that state's gay marriage ban but immediately delayed the implementation of his ruling pending appeals by the state.
The Kentucky ruling came in lawsuits brought by four gay and lesbian couples seeking to force the state to recognize their out-of-state marriages.
The proposed order only requires Kentucky to recognize the marriages of gay and lesbian couples performed in other states or countries. It does not deal with whether the state can be required to issue marriage licenses to same-sex couples.
Martin Cothran, a senior policy analyst for the Family Foundation of Kentucky, criticized Attorney General Jack Conway's handling of the case, accusing him of "spiking" the state's defense by not making persuasive arguments to keep the ban in place.