Anthony claimed that the relationship with the North Carolina teenager "who said she was a high school student" began after she got in touch with him online. "I apologise to everyone I have hurt".
He was released on $150,000 bail and is due back in court September 8 for sentencing, officials said.
Leathers said she encouraged the underage girl to report Weiner to the Federal Bureau of Investigation - a move that resulted in the agency's re-opening of its probe into Hillary Clinton's emails.
Weiner was forced to resign from Congress, where he represented parts of New York City, in June 2011, not long after an explicit picture, sent from his Twitter account, became public. It emerged that at least one photo pictured both Weiner's crotch and Abedin and Weiner's then four-year-old son.
He must register as a sex offender and faces a possible prison term.
As part of his plea, Weiner agreed to forfeit his iPhone.
Aussie coach sees highs and lows in sevens campaign
U.S. would later turn the tables on South Africa , winning 24-12 as both sides eased through from Pool A to the Cup quarters. Scotland sealed back-to-back London Sevens titles as they came from behind to beat England 12-7 at Twickenham .
"I have a sickness, but I do not have an excuse", the former Democratic congressman said.
Weiner's exact sentence will be determined by a judge at a later date.
Although, the unnamed victim never meant to press charges as she thought their convos were consensual.
Huma Abedin filed for divorce from Anthony Weiner May 19, the same day the disgraced congressman pleaded guilty in connection with his sexting case. His sentencing hearing is set for September 8. He later said the emails were duplicates of earlier material already examined by the Federal Bureau of Investigation.
Arlo Devlin-Brown, an attorney at Covington & Burling, said Weiner accepted "full responsibility" for his "inappropriate" actions, adding that Weiner would not further address the matter.
The incident spilled over into the presidential race when investigators seized Weiner's laptop and found a cache of purportedly new emails that Democratic presidential candidate Hillary Clinton sent to Abedin. The emails prompted then-director of the FBI James Comey to announce in October, just prior to Election Day, that the bureau was opening a new inquiry into Clinton's email correspondence policies while she was USA secretary of state.
Clinton's team blamed Comey's decision to release the documents ahead of the presidential voting as the reason for the Democrat's unexpected loss to President Donald Trump.
"While I have provided the Daily Mail with information showing that I have likely been the subject of a hoax, I have no one to blame but me for putting myself in this position", Weiner wrote in a statement to NBC News.