Obama also agreed to Karzai’s demand that American forces abandon small rural outposts that he thinks stoke opposition to his government and its foreign backers and support for the insurgency.

Both steps are crucial to a new peace plan Karzai is pursuing that calls for releasing Taliban prisoners and effectively turning over to the insurgents their strongholds of eastern and southern Afghanistan as part of a political settlement to the war.

“I’ll be going back to Afghanistan this evening to bring to the Afghan people the news of Afghanistan standing shoulder to shoulder with America as a sovereign, independent country,” Karzai said.

For his part, the Afghan president appeared to have accepted Obama’s non-negotiable demand that any U.S. force that remains in Afghanistan after 2014 is subject to the U.S. Uniform Code of Military Justice but receives immunity from Afghan law.

“I can go to the Afghan people and argue for immunity for U.S. troops in Afghanistan in a way that Afghan sovereignty would not be compromised,” said Karzai, who asserted that the immunity issue was as important to the U.S. as “sovereignty and detentions and the continued presence of international forces in Afghan villages” are for Afghans.

The American commander in Afghanistan, Marine Gen. John Allen, reportedly has presented options that call for keeping 6,000 to 20,000 U.S. troops in the country, though administration officials said this week for the first time that the White House might decide not to keep any American soldiers there.

Obama said he and Karzai hadn’t decided on how many U.S. troops might remain, but he underscored that an American mission would be “very specific and very narrow”: training and assisting Afghan forces, and attacking al Qaida and its affiliates.

“That is a very limited mission, and it is not one that would require the same kind of footprint, obviously, that we’ve had over the last 10 years in Afghanistan,” Obama said.

Obama said the pair had discussed the new peace plan Karzai is pursuing with the Taliban, which is envisioned by 2015, and that they’d agreed to try to speed it along with the opening of a Taliban political office in Qatar to facilitate talks.

The new plan calls for Pakistan to replace the U.S. in arranging direct talks between Karzai and leaders of the Taliban-led insurgency, and Obama said the U.S. welcomed the initiative.

He said that after 10 years the U.S. had “come very close to achieving our central goal,” which is to decapitate al Qaida, “to dismantle them, to make sure that they can’t attack us again.”

But in its latest report on the war, the Pentagon said that despite progress in fighting Taliban-led extremists and the training of Afghan security forces, the U.S.-led counterinsurgency campaign "continued to face challenges, including a rise in insider attacks."

"The insurgency’s safe havens in Pakistan, the limited institutional capacity of the Afghan government and endemic corruption remain the greatest risks to long-term stability and sustainable security in Afghanistan," it said. "The insurgents remain resilient and determined, and will likely attempt to regain lost ground and influence through continued assassinations, intimidation, high-profile attacks and the emplacement of improvised explosive devices (IEDs). Widespread corruption continues to limit the effectiveness and legitimacy of the Afghan government."

Karzai pledged that he won’t seek re-election when his second term ends in 2014, saying “the greatest of my achievements” would be an “interference-free election” in which he’d be “very happily, a retired president.”

An Afghan reporter asked Obama what assurances he could give women in Afghanistan that they won’t see their rights restricted by any peace settlement with the Taliban

He replied that while the peace process has to be Afghan-led, the U.S. has made it clear that the Taliban have to recognize the Afghan Constitution and its protections for women.

“The United States strongly believes that Afghanistan cannot succeed unless it gives opportunity to its women,” Obama said. “We believe that about every country in the world.”