But both McConnell and the chairman of the Senate Judiciary Committee, South Carolina Sen. Will Hurd, R-Texas, on the House floor before the vote.
Before joining Mueller's team in May 2017, Weissmann was best known for two assignments: the investigation of now-defunct energy company Enron and organized crime cases when he was a federal prosecutor in Brooklyn.
Most Republicans sided with Democrats on the resolution, though they called it a waste of time.
Though Graham, the Senate Judiciary Committee chairman, said he supported transparency, he asked to amend the resolution to include the appointment of a new special counsel to investigate the FBI's handling of the Hillary Clinton email case and its surveillance of a former Trump adviser.
President Donald Trump has repeatedly called the probe a "hoax" and a "witch hunt".
It called for any report Mueller sends to Barr to be released publicly - except for any classified material.
Those regulations require only that the report explain the decisions to pursue or to decline prosecutions, which could be as simple as a bullet-point list or as detailed as a full-blown report.
Democrats seized on Barr's resistance to making specific promises about the Mueller findings during his confirmation process in the Senate.
Unless Weissmann is leaving for unspecified personal reasons, his departure signals Mueller has likely completed all of the inquiry's major investigative steps, although there could still be additional indictments, said Barbara McQuade, a former senior federal prosecutor in MI.
The resolution easily passed 420-0, despite protests from some Republicans who argued that the legislation was not necessary. He defended Barr, saying he "understands the questions, the turmoil this has caused".
"What happens when it comes back and it says none of this was true - the president did not do anything wrong? Then the meltdown will occur".
Though they voted for the resolution, many Republicans expressed skepticism about the wisdom and likely success of Democrats' quest.
Schumer added that he was "deeply disappointed" in Graham, who chairs the Senate Judiciary Committee.
"I want the American people to know as much as they can and to see as much as they can.it should have been broader", said Rep. Plus, getting the full unclassified report released might be the only way to prevent both sides from selectively leaking about its contents.