A Skadden Arps team led by Gregory B. Craig, a former White House counsel for President Barack Obama, produced a report concluding that, while the trial violated some of Ms. Tymoshenko’s rights, her conviction was supported by the evidence presented at trial. And the report found no evidence that the prosecution was politically motivated.

Mr. Craig, who maintained deep connections to Washington’s Democratic establishment and its press corps, worked to shape the public relations strategy for the release of the report, according to a Justice Department filing released with the settlement.

The filing, which identifies Mr. Craig as “Partner 1” but does not name him, indicates that he arranged for a journalist to receive a copy of the report, then discussed the report with that journalist. The journalist, who is not named in the filing, is David E. Sanger of The New York Times, which published an article in December 2012 about the report quoting Mr. Craig.

Mr. Craig and Skadden Arps should have disclosed that activity under the Foreign Agents Registration Act, known as FARA, which covers both lobbying and public relations on behalf of foreign political interests, the Justice Department said. But Mr. Craig misled both his colleagues at Skadden Arps and officials in the Justice Department’s FARA unit about his interactions with the news media, leading the Justice Department to conclude that the firm was not obligated to register under the act, the settlement filing said.

The settlement “puts law firms on notice that they can’t hide behind their identity as lawyers. If they are doing lobbying work on behalf of foreign countries, they need to register under FARA,” said Rebecca Roiphe, a professor at New York Law School who specializes in legal ethics. “It also shows that the government will not tolerate false statements by lawyers.”

Skadden Arps said in a statement that it had “learned much from this incident” and was “taking steps to prevent anything similar from happening again.”

Mr. Craig’s lawyer declined to comment.

The investigation that led to the settlement with Skadden Arps was handled by the Justice Department’s Counterintelligence and Export Control Section, which includes the FARA unit. But federal prosecutors in the Southern District of New York, to which Mr. Mueller’s team referred illegal lobbying cases, asked witnesses about Mr. Craig’s involvement in Mr. Manafort’s work as recently as last month, according to people familiar with the case.