"After leaving the governor's office I told my wife about this event, which occurred over a decade ago," Schwarzenegger told the Times in a statement that also was sent to The Associated Press early Tuesday. "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.
"I ask that the media respect my wife and children through this extremely difficult time," the statement concluded. "While I deserve your attention and criticism, my family does not."
Schwarzenegger's representatives did not comment further. A spokesman for the former first lady told the Times she had no comment.
The Times did not publish the former staffer's name nor that of her child but said the woman worked for the family for 20 years and retired in January.
In an interview Monday before Schwarzenegger issued his statement, the former staffer said another man -- her husband at the time -- was the child's father. When the Times later informed the woman of the governor's statement, she declined to comment further.
The child was born before Schwarzenegger began his seven-year stint in public office.
Shriver stood by her husband during his 2003 gubernatorial campaign after the Los Angeles Times reported accusations that he had a history of groping women. Schwarzenegger later said he "behaved badly sometimes."
In his first public comments since the couple announced their breakup, Schwarzenegger said last week that he and Shriver "both love each other very much."
"We are very fortunate that we have four extraordinary children and we're taking one day at a time," he said at a Los Angeles event marking Israeli independence. Their children range in age from 13 to 21.
Since his term as California governor ended in early January, Schwarzenegger, 63, has hopscotched around the world, his wife nowhere in sight. While the "Terminator" star appeared confident about the future since exiting politics, cutting movie deals and fashioning himself as a global spokesman for green energy, Shriver, known for her confidence, seemed unsettled.
Shriver, 55, maintained her own identity when her husband entered politics, though she gave up her job at NBC. Their union was often tested in Sacramento, where the former action star contended with a rough seven years of legislative gridlock, a budget crisis and lingering questions about his fidelity.