Barron co-authored a 2011 Justice Department memo that was used to justify the killing of four Americans suspected of terrorism overseas, including the dismissal of a lawsuit filed by the father of cleric Anwar al-Awlaki, who was targeted by a drone strike in Yemen in September of that year.
Earlier this year, a federal court said the CIA, Defense Department and Justice Department did not have to respond to a Freedom of Information Act request to release documents related to the killings, but the ruling was overturned by the Second Circuit Court of Appeals in April.
Liberal and Conservative senators had promised to oppose Barron's nomination to the First Circuit Appeals Court unless the memoirs were made public, and last week, the White House allowed senators to view the memo in a secure room on Capitol Hill.
And Tuesday, the administration said it would not appeal the Second Circuit ruling, and would, in time, release redacted versions of the memo and listings of all classified documents related to the killings. The liberal senators who opposed Barron's nomination without the release of the memo voted to support it Wednesday.
A final vote for confirmation is planned for Thursday afternoon.
While it wasn't clear exactly when the memo will be made public, an official told the Washington Post it would likely be "a matter of weeks."