SAN JOSE, Calif. –  The judge presiding over an intellectual property dispute between Apple Inc. (Nasdaq: AAPL) and Samsung Electronics Co. said the chief executive officers of the contending companies should talk again before the jury begins deliberating.

“I’m going to make one more request that CEOs from both sides speak by phone,” U.S. District Judge Lucy Koh said in federal court today in San Jose, California. “I see risks here for both sides,” she said.

Koh earlier this year ordered Apple CEO Tim Cook to meet face to face with his counterpart at Samsung, Choi Gee Sung. That conference didn’t yield a settlement.

“It’s at least worth one more try,” the judge said today.

Apple sued Samsung in April 2011, accusing it of copying patented designs for mobile devices, and Samsung countersued. The case is the first to go before a federal jury in a battle being waged on four continents for dominance in a smartphone market valued by Bloomberg Industries at $219.1 billion.

Apple is claiming at least $2.5 billion in damages for patent and trade-dress infringement. Cupertino, California-based Apple also wants to make permanent a preliminary ban it won on U.S. sales of a Samsung tablet, and extend the ban to Samsung smartphones.

‘Message Delivered’

Samsung, based in Suwon, South Korea, is trying to persuade the jury to find Apple’s patents invalid and to award unspecified damages for alleged infringement of its patents.

“If all you wanted is to raise that you have IP on these devices, message delivered,” Koh said today. “External valuations” of the intellectual property have been established at the San Jose trial and in other courts, she said.

“In many ways, mission accomplished,” she said. “It’s time for peace.”

Attorneys for both sides said they will obey her request. Apple General Counsel Bruce Sewell, who was in court, told the judge he will deliver the message to his company.

Koh has given each side 25 hours to present its case. The trial is expected to conclude late this month. Jury deliberations may begin as early as Aug. 21, she said.

The case is Apple Inc. v. Samsung Electronics Co. Ltd., 11-cv-01846, U.S. District Court, Northern District of California (San Jose).