Los Angeles (CNN) -- Maria Shriver and her children are "circling the wagons and are working through" the family crisis brought on by the revelation that Arnold Schwarzenegger fathered a child more than 10 years ago outside his marriage, a source close to Shriver told CNN Tuesday.

The former California governor is "way on the outside of the circle" and "rambling around" their huge house with none of his family around, said the source, who asked not to be identified because of the sensitivity of the matter.

The revelation, published first by the Los Angeles Times, came about a week after Schwarzenegger and his wife of 25 years announced that they were "amicably separating."

"After leaving the governor's office I told my wife about this event, which occurred over a decade ago," Schwarzenegger said in a statement obtained by CNN. "I understand and deserve the feelings of anger and disappointment among my friends and family. There are no excuses and I take full responsibility for the hurt I have caused. I have apologized to Maria, my children and my family. I am truly sorry."

Schwarzenegger fathered the child with a household staffer, the Times article said. He has provided support for the child since birth, the source told CNN.

First published by the Los Angeles Times

The staffer told the newspaper that the child was fathered by her then-husband. She declined to comment after learning of Schwarzenegger's statement, the newspaper reported.

The newspaper quoted her as saying she recently retired "on good terms" with the couple after working for them for 20 years.

The newspaper said it was not publishing the former staffer's name or that of her child to protect their privacy.

Schwarzenegger's four children were told about their sibling only recently, with Shriver "methodically" talking to them, the source said. They were "well prepared" before it became public, the source said.

Patrick, Schwarzenegger's 17-year-old son, expressed his feelings Tuesday morning in a Twitter posting:

"some days you feel like s--t, some days you want to quit and just be normal for a bit, yet i love my family till death do us apart. #family"

The oldest child, 21-year-old Katherine, followed with her own tweet later Tuesday:

"This is definitely not easy but I appreciate your love and support as i begin to heal and move forward in life. I will always love my family!"

Shriver, who described her husband as an "A-plus human being" when he was accused of sexual misconduct in 2003, released a written statement Tuesday calling it "a painful and heartbreaking time."

"As a mother, my concern is for the children," Shriver said. "I ask for compassion, respect and privacy as my children and I try to rebuild our lives and heal. I will have no further comment."

People: 'I love my family,' Patrick Schwarzenegger says

Privately with her children, "Maria is very much 'Don't abandon your father,' " the source said "It is very admirable to not show your children the anger and also ask your children to be supportive of their father."

The announcement on May 9 that Schwarzenegger and Shriver were separating did not mention a reason for the split, but said the decision was a mutual one.

"This has been a time of great personal and professional transition for each of us," they said. "After a great deal of thought, reflection, discussion, and prayer, we came to this decision together."

The couple have four children together between the ages of 13 and 21.

After Schwarzenegger and Shriver married in 1986, they became one of America's most famous couples.

He was one of Hollywood's highest-paid actors, and she was a network news anchor and a member of the Kennedy family, America's storied Democratic political dynasty.

Shortly before the 2003 gubernatorial election that catapulted the former actor to political office, however, the Los Angeles Times ran a series of articles in which 16 women accused him of sexually harassing and humiliating them over a 30-year period in incidents on movie sets, at gymnasiums and elsewhere.

One of the women said that when she worked with Schwarzenegger on a movie, he pulled up her shirt to expose her breasts and took a picture, and groped her on two other occasions.

Rhonda Miller filed a lawsuit, saying Schwarzenegger and his campaign staff tried to discredit her by spreading false information about her.

Schwarzenegger said he knew nothing about Miller's allegations and said the timing of the complaints of the 16 women was "rather odd," given that no one had filed a complaint against him in the 35 years that he had lived in the United States.

Schwarzenegger issued a general apology for "behaving badly sometimes" but said many of the allegations were not true "because that's not my behavior."

Shriver took to the campaign trail to defend her husband a day after the allegations surfaced and just four days before the 2003 recall election.

"You can listen to all the negativity, and you can listen to people who have never met Arnold, or who met him for five seconds 30 years ago. Or you can listen to me," she said in a speech to a Republican women's group in Orange County.

"I wouldn't be standing here if this man weren't an A-plus human being. I wouldn't be taking my time, I wouldn't have left my job that I love, I wouldn't be doing any of this if I didn't believe in this man."

Speaking to reporters after the speech, Shriver described her husband as "an extraordinary man" and took issue with the characterization that he behaves boorishly toward women.

"He's honest, he's sensitive, he's sincere. And he is gracious with every bone in his body," she said. "I have met thousands and thousands of women who have come up to me and said they have worked with him, they have worked for him, alongside (him) and he has been an extraordinary gentleman."

She also said she thinks it took "tremendous courage to stand up and say, 'I'm sorry if I offended anybody. I apologize. That was not my intention.' "

Shriver said then that she was not personally hurt by the allegations.

"Nothing hurts, because I know the man that I'm married to," she said. "Has he said and done everything absolutely (right) 100% of the time? No. But I don't know anybody on the planet, male or female, who's led a perfect life.

"I think it's really important to understand who Arnold is today, and I think it's taken great courage on his part to get into this race."

Schwarzenegger, a naturalized U.S. citizen from Austria who gained fame as the youngest winner of the Mr. Universe bodybuilding contest, was governor until January. He has been busy in recent months reviving his acting career and signing movie deals -- including a project to do another installment of his "Terminator" series.