Still, the government relied on Mr. Rechnitz’s testimony to build a narrative that ultimately led to Mr. Seabrook’s conviction. Mr. Rechnitz, who pleaded guilty to fraud conspiracy and is cooperating with the government, said he brokered a deal between Mr. Seabrook and Murray Huberfeld, a family friend and the founder of the hedge fund Platinum Partners. Mr. Seabrook agreed to invest the union’s retirement money in Mr. Huberfeld’s fund, and in exchange, Mr. Huberfeld would pay him a cut of the profit, the government said.

Mr. Seabrook ended up investing $20 million in the fund, $19 million of which the union eventually lost.

It was a risky proposition for the prosecution to once again depend on Mr. Rechnitz, whose inconsistent testimony in the last trial seemed to lead to the hung jury. A juror in the fall referred to him as a “straight-up liar.”

The government said that the bribe was delivered in a nearly cinematic scene: One December evening, Mr. Rechnitz took $60,000 in cash from a safe in his office and crammed the bills into a handbag he had purchased from the Salvatore Ferragamo store, Mr. Seabrook’s favorite designer. He then delivered the bag to Mr. Seabrook before they attended a Torah dedication ceremony together.

Mr. Huberfeld, who was Mr. Seabrook’s co-defendant in the first trial, pleaded guilty to wire fraud conspiracy in May.

In this trial, Mr. Rechnitz was caught in fewer inconsistencies on the stand, and he seemed more prepared to answer the defense’s questions. He also spent less time talking about his relationship with Mr. de Blasio, testimony that dominated much of the first trial.

At that time, he said he had donated tens of thousands of dollars to Mr. de Blasio’s re-election campaign in exchange for a direct line to City Hall, including the mayor’s personal phone number and email address. During the second trial, prosecutors kept Mr. Rechnitz’s testimony more narrowly focused on his shady business dealings and on the payment to Mr. Seabrook.