"I've been unfaithful to my wife," he said, adding that he had spent the last five days "crying in Argentina".
Mr Sanford said what he had done was wrong. "Period. End of story."
He said his wife and family had known of the affair for the past five months and that he was resigning his post of chair of the RGA.
Mr Sanford was missing for several days from late last week, with attempts to establish his whereabouts making headline news.
His staff had said he was walking on the Appalachian Trail in the east of the US, while his wife said she did not know where he had gone.
On Wednesday, Mr Sanford admitted he had been on a private visit to Argentina because he had "wanted to do something exotic".
I developed a relationship with... a dear, dear friend from Argentina and as a consequence I hurt her, I hurt you all, I hurt my wife, I hurt my boys, I hurt a lot of different folks
He said he had not ordered his staff to cover for him, allowing them to continue in the belief had gone hiking.
"I let them down by creating a fiction with regard to where I was going," Mr Sanford said.
A South Carolina newspaper, The State, has published excerpts of e-mails between Mr Sanford and the woman.
The governor's office said it would not dispute the authenticity of the exchanges.
One from Mr Sanford said:" You are special and unique and fabulous in a whole host of ways that are worth a much longer conversation."
In another, he writes: "In the meantime, please sleep soundly knowing that despite the best efforts of my head my heart cries out for you, your voice, your body, the touch of your lips, the touch of your finger tips and an even deeper connection to your soul."
The BBC's Justin Webb in Washington says Mr Sanford is known for enjoying spending time on his own.
But his decision to leave South Carolina without a leader for several days had left many questioning his judgement and may have cost him his future career, our correspondent says.
Mr Sanford has more than a year left in his second term as governor and is constitutionally barred from standing for office again, but has not said whether he will stand down.
The governor had battled the Obama administration but was forced by a court ruling earlier this year to accept $700m in stimulus money from the federal government.
"This decision is terrible news for every taxpayer in South Carolina, and even more so for future taxpayers who will ultimately bear the responsibility of paying for this so-called 'stimulus' without seeing any benefit from it," he said in a statement at the time.
Mr Sanford's resignation is the latest problem to beset the Republican party.
Last week, Republican Senator John Ensign of Nevada admitted to an extra-marital affair with a member of his campaign staff.
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