"Their marriage continues to be quite healthy," he says of the Live! star and Mark Consuelos

Kelly Ripa and Mark Consuelos
Credit: Dimitrios Kambouris/WireImage

Kelly Ripa‘s rep has a reply to the claim that the star’s 12-year marriage to actor Mark Consuelos is over: No way.

“There is no truth to the story. Their marriage continues to be quite healthy, and the National Enquirer should be ashamed for fabricating such an untruthful story,” Ripa’s spokesman, Stan Rosenfield, told PEOPLE Wednesday, after the paper reported the couple have split.

“Their story is 100 percent false,” Rosenfield added.

The couple’s lawyer, Michael Zweig, has demanded a “full public retraction” of the story and an apology from the paper.

On Monday, Ripa and Consuelos walked hand-in-hand down the red carpet at the New York premiere of the new movie Cadillac Records.

Regis Philbin’s on-air cohost of Live! since February 2001, Ripa, 38, has been wed to Conseulos, 37, since 1996, after having met the year before while both were acting on the daytime drama All My Children. Together they have three children: Michael, 11; Lola, 7; and Joaquin, 5.

Last year, Ripa said of Consuelos: “I’ve had a crush on my husband since the day I met him.”