Story highlights The Supreme Court says a Kentucky clerk who refused to issue marriage licenses on religious grounds must start doing so

The order marks the first time the issue of same-sex marriage has come back to the Supreme Court since the June landmark ruling

A Kentucky judge has set a hearing for Thursday for the clerk Kim Davis

Washington (CNN) The ACLU of Kentucky filed a motion in federal court Tuesday asking a judge to hold county clerk Kim Davis in contempt of court for failing to comply with the Supreme Court and issue marriage licenses to same-sex couples.

In court papers, the ACLU lawyers note that following the Supreme Court's order late Monday night, the plaintiffs in their case went to the Rowan County Clerk's office at 8 a.m. Tuesday to obtain a marriage license.

"Unfortunately , they were again denied by a deputy clerk who asserted that no marriage licenses would be issued," the ACLU said in a statement. The lawyers argue that Davis has "violated a definite and specific order" and that the Court is left "with no choice but to hold her in contempt."

Davis, of the Rowan County Clerk's office, has refused to issue any marriage licenses since the U.S. Supreme Court decision -- Obergefell v. Hodges -- came down. She is an Apostolic Christian who says that she has a sincere religious objection to same-sex marriage. Other clerks in the state have expressed concern, but Davis is the only one turning away eligible couples.

The ACLU said they do not compel her compliance through incarceration, but instead they urge the court to impose financial penalties "sufficiently serious" to compel her immediate compliance without further delay.

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