The former governor of the state of California made the admission just a week after he and wife Maria Shriver announced their separation after 25 years of marriage. According to reports, Shriver decided to move out of their Brentwood mansion earlier this year after discovering the news.
"After leaving the governor's office I told my wife about this event, which occurred over a decade ago," Schwarzenegger told the Los Angeles Times in a statement.
"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 apologised to Maria, my children and my family. I am truly sorry."
He continued: "I ask that the media respect my wife and children through this extremely difficult time. While I deserve your attention and criticism, my family does not."
Representatives for Schwarzenegger and Shriver declined to make any further comments.
The child's mother is alleged to have worked for the family for 20 years before her retirement in January. Prior to his confirmation, the unidentified female had insisted that the father was another man. When informed of Schwarzenegger's decision to release the statement, she is said to have declined to comment.
Schwarzenegger has said that he and former wife Shriver, who have four children together, still "love each other very much".
> Arnold Schwarzenegger 'mistress' speaks out