On Thursday morning, prosecutors dropped their objection to a "special master" being appointed to further insulate sensitive files from possible undue disclosure to investigators.
Jones, who was a federal and state prosecutor, was appointed special master by Judge Kimba Wood at a standing-room-only hearing in U.S. District Court in Manhattan. Cohen was present but said nothing.
Wood said that letters recently filed by lawyers in the case "convinced me that this process" of reviewing the seized files "can go quickly with a special master."
Those files are currently being electronically scanned by authorities but have not actually been looked at by the investigators.
FBI agents on April 9 seized a number of electronic devices and paper documents from Cohen, including four phones and an iPad. One of the iPhones seized will take an estimated 104 days to extract files from, according to prosecutor Tom McKay.
Jones has experience acting as both a mediator and as a court-appointed monitor, but she was not one of the seven names recommended by either the prosecution or Cohen's lawyers as special master for Cohen's case. Her payment for her work has yet to be decided. Jones has told Wood that she would commit 90 percent of her work week to reviewing Cohen's files.
Jones served 16 years as a judge in that Manhattan court after being appointed by President Bill Clinton in 1995.
She previously worked as a prosecutor in the Office of the United States Attorney for the Southern District of New York — the same office now investigating Cohen.
Jones' boss during her tenure as a federal prosecutor was Rudy Giuliani, the former New York mayor who was recently named to the legal team of Trump to represent him in an investigation by special counsel Robert Mueller.
Jones joined the Bracewell law firm in May 2016, shortly after Giuliani left as a name partner in that firm.
The files seized by Cohen include ones related to a $130,000 payment he made to porn star Stormy Daniels right before the 2016 presidential election. Daniels has said that payment was in exchange for her silence about an alleged affair she had with Trump in 2006. The White House denies such an affair.
Daniels' lawyer Michael Avenatti earlier Thursday had asked Wood to allow him to formally participate in the case. He wants Daniels to be allowed to assert claims of attorney-client privilege that might exist in Cohen's files because her previous attorney in the matter, Keith Davidson, negotiated the nondisclosure agreement with Cohen.