'RHONY' Star Carole Radziwill Allegedly Recorded Private Convos, Woman Wants It Off the Show

1:20 PM PT -- Bravo says the judge just ruled in its favor, rejecting Barbara's motion to block ... which means Barbara stays in the picture ... TV show, in this case.

'Real Housewives of New York' star Carole Radziwill secretly recorded talks she had with a woman, and their private convos will air Wednesday night unless a judge steps in ... according to new legal docs.

Barbara K, founder of a DIY company, claims she had what she believed to be a private discussion with Carole at a charity party at Bagatelle in Manhattan. She says 'Real Housewives' was filming ... but their convo took place far away from cameras and she didn't know Radziwill was mic'd.

According to the docs ... Barbara had a phone call with Carole more than a month later that was also recorded without her knowledge. She claims Radziwill called her back afterward to inform her both interactions had been recorded for 'RHONY.'

Barbara says she told Carole she didn't have permission to use the convos on the show or anywhere else, and her lawyer fired off a letter to Bravo telling them the same.

Despite this, 'RHONY' still plans to put Barbara in Wednesday night's episode ... which is why she requested an emergency court order to stop the show from using her likeness and/or voice.