Weiner, who was forced to resign from Congress after his proclivity for sexting was made public, acknowledged he continued to exchange "inappropriate messages" with strangers as recently as last summer.
Weiner has admitted sending lewd pictures of himself to young women.
Screenshots of Weiner's text messages in August 2012 were posted Monday on The Dirty website, which said it obtained the pictures from a source.
"I just want to clarify that although I was 22 and technically an adult, I was obviously immature and I acted irresponsibly. I realize my correspondence with Mr. Anthony Weiner was a huge mistake and I am embarrassed by it," the woman said, telling The Dirty the relationship dropped off last November with only a single contact in December and another in April.
Weiner, who was accompanied by his wife, Huma Abedin, acknowledged at a late-afternoon news conference his behavior was wrong but it is "behind me." He apologized to his wife and to the women to whom he sent the pictures.
"I want to reiterate how sorry I am I did these things," he said.
Weiner said he continued to send the pictures after he resigned from Congress and while he and his wife were going through counseling.
"That was also the time my wife and I were working through some things in our marriage," he said, adding with just 49 days to go before the New York mayoral primary, he's "surprised things didn't come out sooner." At the same time, he said, "Things are not that different than yesterday."
Asked how he would respond to opponents who were calling for him to drop out if the race, Weiner said, "I'm sure many of my opponents would like me to drop out of the race." But he said he hoped New Yorkers would give him a second chance.
Weiner said some of the things posted by The Dirty were untrue but he wasn't going to argue about the specifics.
Abedin said her husband has made "some horrible mistakes ... [but I] strongly believe it is between us and our marriage. ...
"I love him. I have forgiven him. I believe in him. ... We are moving forward."