Singh, 67, posted two tweets, acknowledging the relationship with the Rajya Sabha TV anchor.
"I have no hesitation in accepting my relationship with Amrita Rai. She and her husband have already filed a mutual consent divorce case."
"Once that is decided we would formalise it. But I do condemn encroachment in our private life," he said.
Rai separately posted: "I have separated from my husband and we have filed a mutual consent divorce papers. After which I have decided to marry with Digvijaya Singh."
Twitteratti posted comments, mostly snide ones, about the couple coming out in the open about their relationship.
Singh's wife had passed away in the past year after battling cancer.
Pictures of the couple in an intimate moment went viral on the net.