How Catherine Zeta-Jones and Michael Douglas Are Fighting to Save Their Marriage

09/25/2013 at 08:00 AM EDT

How Catherine Zeta-Jones and Michael Douglas Are Fighting to Save Their Marriage
Michael Douglas and Catherine Zeta-Jones
The love story is not over between Catherine Zeta-Jones and Michael Douglas.

In the four months since the Oscar-winning couple decided to separate, they have wrestled with whether their almost 13-year marriage can – and should – be saved. And for now, neither star is ready to call it quits.

"They're not giving up," a source close to Zeta-Jones tells PEOPLE in this week's cover story.

While they continue to lead separate lives – Zeta-Jones, 44, was in China at a promotional event when Douglas, 69, won his Emmy for Behind the Candelabra on Sunday – they are still speaking multiple times each day, says the source.

The couple, who traditionally celebrate their shared Sept. 25 birthday together, are expected to be apart, with Zeta-Jones remaining in Bedford, N.Y., with their children Dylan, 13, and Carys, 10, and Douglas in New Mexico shooting his upcoming thriller The Reach.

Speaking to PEOPLE at the Emmys, the actor reaffirmed his commitment to his marriage.

"We're working things out, talking," he said, "and we'll see how she goes."

For much more on Michael and Catherine's marriage struggle, pick up this week's PEOPLE, on newsstands Friday

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