"The United States has assured us of continued support and assistance on all matters," Karzai said at a joint news conference with Defense Secretary Donald H. Rumsfeld.
Rumsfeld, who flew here from Pakistan where he toured U.S. humanitarian assistance to earthquake victims, told reporters the United States is not going to abandon Afghanistan.
"We certainly remain committed to our long-term relationship," Rumsfeld said.
Asked about the Pentagon's announcement Tuesday that U.S. troop strength in Afghanistan will be reduced from about 19,000 to about 16,000 next year, Karzai said this was a matter for the United States to decide.
"I don't think it will have an effect on the ground," he said.
On his way to Chaklala Air Base in Pakistan, Rumsfeld told reporters it appears unlikely that Osama bin Laden, if still alive, is in full command of the Al Qaeda terrorist network.
Rumsfeld said the U.S. government does not know bin Laden's whereabouts but it's a "reasonable assumption" that the Saudi exile is in the remote area along the Afghan-Pakistan border.
Rumsfeld flew to Pakistan on Tuesday for an unannounced visit to the areas hardest hit by the Oct. 8 earthquake that killed an estimated 87,000 people, forced at least 3 million from their homes and triggered the deployment of hundreds of U.S. troops to provide medical, logistical and other assistance.
Vice President Dick Cheney also visited Pakistan this week.
In an interview aboard an Air Force C-32 flight from Washington, Rumsfeld said he does not know where bin Laden is, or even whether he's still alive. But a good guess is that he's hiding along the border.
"It's a reasonable assumption, if you don't know where he is, but if you want to guess, he has in the past operated in those areas," Rumsfeld said.
Speaking earlier in Washington before he departed for Pakistan, Rumsfeld said a reduction in the size of the U.S. military force in Afghanistan — from about 19,000 to about 16,000 — would not diminish the hunt for bin Laden and other terrorists in the area along the Afghan-Pakistan border.
Rumsfeld told reporters he wanted to visit Pakistan to see the humanitarian relief operations and to demonstrate U.S. support for Pakistan and its recovery efforts. He added that he found it interesting that bin Laden has not been heard from publicly in nearly a year.
"I don't know what it means," Rumsfeld said. "I suspect that in any event, if he's alive and functioning that he's probably spending a major fraction of his time trying to avoid getting caught. I have trouble believing that he's able to operate sufficiently to be in a position of major command over a worldwide al-Qaida operation, but I could be wrong. We just don't know."
U.S. military aircraft have dropped millions of pounds of relief supplies to the areas hardest hit by the Oct. 8 quake. The U.N. humanitarian chief, Jan Egeland, said last week that there were hundreds of thousands of lives at stake as winter weather descends on northwestern Pakistan and the disputed Kashmir region.
Many uprooted survivors of the quake have been living in hundreds of spontaneous and organized refugee camps, many at elevations above 5,000 feet.
The U.S. military has about 850 people assisting, including air crews, the 212th Mobile Army Surgical Hospital, the 3rd Medical Battalion, Naval Mobile Construction Battalion 74 and others.
At Qasim Air Base, where U.S. forces maintain helicopters used in disaster relief flights, Navy Rear Adm. Michael LeFever, the senior U.S. commander of the humanitarian mission, told reporters Wednesday that the presence of American troops in Pakistan had helped change perceptions of Westerners in the area.
"We've become a symbol of hope and relief," he said.
LeFever mentioned that one of the popular toys for children in the area now was a model of a Chinook helicopter. People have come to associate American choppers with relief, he added.