For the companies to win, their lawyers have to convince Judge Kessler that the companies are model citizens, incapable of committing bad acts.
"This is a golden opportunity for this particular person, a person who may not have many years left," said Judge Kessler.
"This case," he told Judge Kessler, "is in trouble, serious trouble."
Early in the case, Judge Kessler ruled that the government could seek $280 billion from the companies as a remedy for any past fraudulent acts.
To speed the case along, Judge Kessler has instructed each side to conduct direct examinations of witnesses outside court before their appearances.
But after an outcry from fishermen and state officials, Judge Kessler imposed a less stringent compromise.
In October 1999, Judge Kessler ordered the preservation of all documents and records that could be related to the lawsuit.
The government lawyers were not deterred by Judge Kessler's expressions of concern.
The meeting began with both sides sitting before Judge Kessler for about 25 minutes.
That is the category most directly affected by Judge Kessler's order.