The revelation came Tuesday, just a week after Schwarzenegger and wife Maria Shriver announced they had separated.
Schwarzenegger reportedly provided financial support for the child, only revealing the infidelity to his wife after leaving office in January. The couple had been married 25 years and has four children together.
In a statement, Schwarzenegger said he was "truly sorry" for his actions.
"After leaving the governor's office I told my wife about this event, which occurred over a decade ago," Schwarzenegger said. "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."
Hours later, Shriver also released a statement.
"This is a painful and heartbreaking time," she said. "As a mother, my concern is for the children. I ask for compassion, respect and privacy as my children and I try to rebuild our lives and heal."
Schwarzenegger's one-time mistress was married at the time of the affair and passed the baby off as her husband's child.
The unidentified staff member retired in January before the revelation of the affair after working for the family for 20 years. She reportedly left her position voluntarily and "on good terms."
According to the New York Times, the child is a 14-year-old boy. The mother was reportedly working around the house pregnant at the same time Shriver was pregnant with the youngest of the couple's four children, Christopher, who is now 13.
The woman and child did not live in the house.
After news of the affair, their son Patrick Schwarzenegger tweeted, "Some days you want to quit and just be normal for a bit, yet I love my family, till death do us apart," referencing a popular song by Fort Minor. The song is about the consequences of choosing a career over family.
The couple's oldest daughter Katherine also tweeted, "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!"
When Schwarzenegger first began his run for governor, there were allegations he had groped and harassed up to a dozen women. At first he denied, then admitted to them in interviews during his campaign.
"I keep saying it now and I said it yesterday, I'm sure that some of it was true. Many of the stories I can't recall," Schwarzenegger said.
Shriver stood up for him and helped get him elected governor a few months later.
"I'm not going to go down that road," she responded to questions from members of the media. "I don't believe in gutter politics and I don't believe in gutter journalism."
In an Exclusive Eyewitness news poll conducted by SurveyUSA, respondents were asked whether Schwarzenegger's image will forever be tarnished or can it be repaired. Fourty-four percent said it would forever be tarnished, 40 percent said it can be repaired and 16 percent said they were not sure.
How Schwarzenegger likely kept his child a secret
Despite being one of the most recognizable people on the planet, a two-term governor and a world champion bodybuilder, Schwarzenegger managed to keep an incredible secret from the public and his wife.
Mark Vincent Kaplan, a prominent attorney who has represented celebrities in splits like Kevin Federline with Britney Spears, says the strongest incentive to maintain secrecy is the welfare of the innocent child, especially if the child never knew who their real dad was.
"This child has grown up probably believing that his mother's spouse is his father and it's really a difficult thing all of a sudden for this child to deal with this type of information," Kaplan said.
To minimize the chance of a leak, parties seek secret contracts, written or verbal, which the courts never see.
A contract can determine custody and defines the manner in which financial assistance will be provided over time.
Since child support is already required by law, additional financial support may be provided for the mother on the condition she keeps the relationship a secret. The objective is to keep everyone happy for as long as possible, Kaplan said.
"You want to be able to make sure there's an incentive for someone not to have to ask for the assistance of the court," Kaplan said.
It is likely that the child will learn his or her paternity at some point. A child has rights to access their biological information based on medical need.
Politicos react to Schwarzenegger's secret
In a building where secrets are tough to keep, the former governor managed to keep his secret from getting out in Sacramento.
Senate President Darrell Steinberg (D-Sacramento) spent a lot of time with Schwarzenegger haggling over the budget and he said he didn't know.
"It is hard to keep a secret but no, I had absolutely no idea," Steinberg said.
But the expensive aides Schwarzenegger surrounded himself with had an inkling some of the infidelity rumors were true, especially after several women the actor worked with on movie sets accused him of groping them in the weeks before being elected in 2003.
"It's appalling but not at all surprising," a former aide told Eyewitness News. "None of the political people who were close to Arnold ever believed all his denials about this stuff. I wish I could say I am surprised. The heartbreaking part is that I'm sure Maria was."
Sean Walsh, a former advisor to Schwarzenegger, was also caught by surprise by the news.
"From his Hollywood life, I don't see this as a surprise," Walsh said. "Were this to occur when he was governor, I would be shocked and I don't believe some of the other rumors that were floating around when he was governor."
Shriver joins a long list of political wives publicly embarrassed by cheating husbands. Some political watchers think this revelation further ostracizes him from his party.
"If you're a Republican already being questioned by social conservatives as to whether you're a RINO, or a real Republican, this kind of scandal only reinforces the point that you're not one of them," said Dr. Bruce Cain of the University of California's Washington Center.
Social Conservative groups say this scandal will ruin Schwarzenegger's political legacy.
"I'm glad the woman chose life, but it's unfortunate Arnold would take advantage of someone working for him in this kind of situation. It's just sad," said Karen England of the Capitol Resource Institute.