Offending couples could face $10,000 fines, nine months in prison or both.
"To have a marriage that you know is illegal where you reside, and you have the intent of coming back to that state where its illegal, that's a defrauding of the government," said the Family Council Julaine Appling.
"[This] law has been around since 1915, which was five years before women got the right to vote," said Civil Rights advocate Glenn Carlson.
Prosecution of the law in any same-sex marriage cases would be up to the discretion of individual district attorneys.