"It is with hearts full of sadness that we have decided to separate," the parents of daughter Apple, 9½, and son Moses, 7½, announced on the GOOP website under the heading "Conscious Uncoupling."
Paltrow, 41, PEOPLE's 2013 Most Beautiful Woman, and Coldplay frontman Martin, 37, say they had been "working hard for well over a year," both together and "some of it separated," to make their relationship work.
"We have come to the conclusion that while we love each other very much we will remain separate," they say. "We are, however, and always will be a family, and in many ways we are closer than we have ever been."
Saying they are "parents first and foremost," they called on the public to respect "our space and privacy to be respected at this difficult time."
"We have always conducted our relationship privately, and we hope that as we consciously uncouple and co-parent, we will be able to continue in the same manner," concludes the announcement, signed, "Love, Gwyneth & Chris."
The couple tied the knot in a secret ceremony in late 2003 and for years have faced breakup rumors, which as long ago as 2008 Paltrow called "ridiculous."