The Trump Organization, which operates hotels, golf courses and other real estate around the world, could face up to $1.6 million in fines for the three tax fraud counts and six other counts it faces, if convicted.
Trump, a Republican who last week launched another bid for the presidency in 2024, has called the charges politically motivated.
The district attorney's office charged the Trump Organization and Weisselberg, who pleaded guilty to charges including grand larceny and tax fraud in an August deal with prosecutors calling for a jail sentence of five months if he testified truthfully in the trial.
Weisselberg, who has worked for the company for nearly half a century, is on paid leave from the Trump Organization.
U.S. Attorney General Merrick Garland on Friday named a special counsel to oversee the Justice Department's investigations related to Trump including his handling of sensitive government documents after leaving office and efforts to overturn the 2020 election.