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Secretary of State Hillary Clinton apologized Tuesday for the NATO airstrike that killed 24 Pakistani soldiers last year -- and announced that Pakistan has in turn agreed to re-open supply lines into Afghanistan that have been closed since the incident.

The deadly airstrike last November crippled already-damaged relations between the United States and Pakistan. Islamabad retaliated by shutting down vital supply lines into Afghanistan, and the two sides had been struggling ever since to reach a deal -- with a U.S. apology said to be a key demand of the Pakistanis.

The Obama administration resisted until now. Clinton, for the first time, delivered that apology Tuesday morning during a phone call with Pakistani Foreign Minister Hina Rabbani Khar.

"We are sorry for the losses suffered by the Pakistani military," Clinton said in a statement.

She said that during the phone call, she "once again reiterated our deepest regrets for the tragic incident" last November, and "I offered our sincere condolences to the families of the Pakistani soldiers who lost their lives."

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Clinton added: "We are committed to working closely with Pakistan and Afghanistan to prevent this from ever happening again."

The Defense and State departments confirmed that Pakistan has in turn agreed to re-open the shuttered supply routes. Clinton said in her statement she was "pleased" to announce the opening, and noted Pakistan would not charge a transit fee, which had been a source of contention.

"This is a tangible demonstration of Pakistan's support for a secure, peaceful, and prosperous Afghanistan and our shared objectives in the region," Clinton said.

The statement followed signs that such a deal was near. Gen. John Allen, the U.S. commander in Afghanistan, had visited Islamabad twice over the last week to help seal the deal.

Pentagon spokesman Capt. John Kirby told Fox News that no vehicles or convoys have started moving across the border yet but confirmed the deal. The closure since November has cost the U.S. more than $2 billion.

The first trucks carrying NATO goods should move across the border on Wednesday, sources said, but it could take days to ramp up supplies to pre-attack levels.

Pakistan's ambassador to Washington, Sherry Rehman, released a statement praising Clinton later in the day Tuesday. "We appreciate Secretary Clinton's statement, and hope that bilateral ties can move to a better place from here. I am confident that both countries can agree on many critical issues, especially on bringing peace to the region," Rehman said.

The dispute over the supply lines had plunged relations between Pakistan and the U.S. to new lows, coming after a CIA contractor killed two Pakistanis and the unilateral U.S. raid on Usama bin Laden's Pakistani compound. Tensions are compounded by the U.S. suspicion that Pakistan supports the Taliban -- Pakistan also angered U.S. lawmakers when it sentenced to decades in prison the Pakistani doctor who helped the U.S. track bin Laden.

With the supply lines closed, the U.S. has been forced to use more costly transportation routes through Russia and Central Asia. Defense Secretary Leon Panetta has estimated the cost at an extra $100 million a month. He warned that it could get more expensive as the U.S. starts to withdraw equipment in advance of the 2014 troop drawdown in Afghanistan.

The Associated Press contributed to this report.