“What I was trying to say  that BP understands how deeply this affects the lives of people who live along the gulf and depend on it for their livelihood  will best be conveyed not by any words but by the work we do to put things right for the families and businesses who’ve been hurt,” he said in a statement.

Image President Obama said that in talks with Mr. Svanberg, he had emphasized the desperation of many people on the Gulf Coast. Credit... Charles Dharapak/Associated Press

Mr. Svanberg said the BP board, which met in emergency session on Monday in advance of the White House meeting, had agreed not to pay further dividends to shareholders this year. Faced with mounting criticism of his company, including from within the oil industry, he denied reports that BP had taken safety shortcuts on the Deepwater Horizon rig, where an April 20 explosion killed 11 workers and set in motion the leak that Mr. Obama has called the worst environmental disaster in American history.

Still, Mr. Svanberg said he wanted to “apologize to the American people on behalf of all the employees of BP.”

The resolution of both the $20 billion fund, to be held in escrow, and the looming controversy over as much as $10.5 billion in dividends BP had been prepared to pay out provided the first triumphal moment for Mr. Obama since the disaster. Weeks of bad news about failed attempts to plug the well and ever-increasing estimates of how much oil was leaking were creating a political crisis for a president who had promised to restore competence to government.

The agreement, by settling much uncertainty about BP’s ultimate liabilities, also gave the company a boost on Wall Street. After the announcement, markets regained ground lost earlier in the day and finished the day mixed. BP shares, which had been down as much as 4.6 percent, closed up 1.4 percent, or 45 cents, to $31.85.

Mr. Obama’s announcement and subsequent information from the White House and BP was in effect a response to some critics, including normally supportive commentators, who had faulted him the night before for being short on specifics about the disaster in his first address to the nation from the Oval Office.

Nonetheless, as the administration also disclosed this week, new government reports suggest that oil is gushing at far greater volume than many had thought, as much as 60,000 barrels a day. BP does not yet have the capacity to capture that much oil in ships stationed above the leak, a mile down on the ocean floor.