In New York’s ornate Capitol building in Albany, a plan two years in the making was taking root. Everyone would benefit: Victims of mesothelioma, a rare but deadly form of cancer caused by exposure to asbestos, would be sent to a reputable law firm; the firm would pick up new clients; and a well-regarded cancer research clinic would receive funds.

And at the center of all this was Sheldon Silver, then the State Assembly speaker, prosecutors say.

The alleged arrangement, which has become the heart of the corruption case against Mr. Silver, came into focus at his trial on Wednesday, as Dr. Robert N. Taub of Columbia University testified in federal court in Manhattan as a government witness. Prosecutors charge that Mr. Silver, a Democrat from the Lower East Side, traded official actions for $3 million in an illegal kickback scheme.

Dr. Taub, who testified under a nonprosecution agreement, ran a clinic at Columbia dedicated to mesothelioma research. The clinic long relied on government grants, wealthy donors and even gifts from law firms and their foundations to fund his research.

But Weitz & Luxenberg, a major personal injury law firm that represents victims of mesothelioma, had not been a donor to Dr. Taub’s research, and he testified on Wednesday that he wanted to change that.