The case was among a number of political corruption cases that were overturned after the United States Supreme Court in 2016 narrowed the activity that could constitute corruption. In a ruling that reversed the conviction of Bob McDonnell, a former Republican governor of Virginia, the court found that such activity must involve concrete and formal government decisions or actions, and not mere political courtesies like setting up a meeting.

Mr. Silver’s retrial was widely watched as a test of the government’s ability to prosecute official corruption under the narrower definition. But from the outset, prosecutors this time seemed to hold an upper hand: The original verdict was only thrown out because the judge’s jury instructions were too broad, as defined by the Supreme Court decision.

The retrial in Federal District Court in Manhattan seemed to move at double speed: Instead of stretching over one month, as the first trial had, the second trial was completed in two weeks, as prosecutors quickly made their case that Mr. Silver, 74, had obtained the illicit payments. In return, prosecutors said, Mr. Silver took a series of official actions that benefited a cancer researcher at Columbia University and two real estate developers in New York.

“Sheldon Silver repeatedly used his enormous public power for his own enormous private gain,” a prosecutor, Tatiana R. Martins, told the jury in a closing statement on Thursday.

Mr. Silver sat impassively as the jury forewoman announced the verdict, which came at the end of the first full day of deliberations. His lawyer, Michael S. Feldberg, said he planned to appeal, citing “multiple legal issues.”