* Clinton and Lavrov call talks positive and constructive

* Agreement to work for new strategic arms pact

* Common interests on N. Korea, Iran and Afghanistan

(Recasts with background)

By Sue Pleming

GENEVA, March 6 (Reuters) - The United States and Russia pledged on Friday to rebuild their strained relations and launched a plan to reach agreement on strategic nuclear missiles by the end of the year.

U.S. Secretary of State Hillary Clinton and Russian Foreign Minister Sergei Lavrov, meeting in Geneva, agreed to find common approaches on Iran, Afghanistan and North Korea.

They discussed areas of disagreement, too, from human rights to Georgia, where Russia fought a short war in 2008 over a breakaway region, and Kosovo, whose independence is recognised by the West but not by Russia.

Clinton said after the meeting it would take time to rebuild the relationship and this would require "more trust, predictability and progress".

Their differences did not overshadow the talks, which marked a further stage in President Barack Obama's efforts to reach out to other countries and make U.S. diplomacy more effective.

"We exchanged our vision of immediate priorities in our relations. I am convinced, and the secretary of state will share my opinion, these priorities largely coincide," Lavrov told a news conference.

Symbolising Obama's intention to improve relations, Clinton presented Lavrov with a small box bearing a red button marked "reset", and the overall tone was conciliatory, with Clinton and Lavrov joking with each other on first-name terms.

"This is a fresh start not only to improve our bilateral relationship but to lead the world in important areas," Clinton said.

Lavrov said Russia and the United States would work honestly and openly together even on areas of difference.

"We understood that our bilateral relations are getting a chance today we cannot afford to miss," he said.

The tone was in contrast to relations under Obama's predecessor George W. Bush, strained by issues including Russia's 2008 military intervention in Georgia and U.S. plans to build a missile interception system in eastern Europe.

Both ministers agreed to work together on a range of issues. Clinton said the priority was a new treaty on strategic arms.

MOST SERIOUS THREAT

The United States wanted to reach agreement by the end of this year, when the Strategic Arms Reduction Treaty (START I) expires, she said. Lavrov called the 1991 pact "obsolete".

Clinton said they wanted to present a plan to Obama and Russian President Dmitry Medvedev before they meet at the G20 summit in London on April 2, so they could agree on instructions to be given to negotiators.

Besides trying to reduce their own arsenals, the two powers agreed to work together to prevent nuclear weapons spreading.

"The most serious threat facing humanity ... is a potential nuclear weapon in the hands of an irresponsible actor. Both Russia and America know that we have to work together to try to prevent that," Clinton said.

Lavrov said Moscow and Washington would try to reach agreement soon on how to deal with the nuclear programmes of Iran and North Korea.

He also sought to assuage U.S. concerns about Iran's interest in buying Russian S-300 air defence systems, which could help repel possible Israeli or U.S. air strikes.

Clinton said the United States was reviewing its policy towards Iran and would welcome Russian advice on how to prevent Tehran from acquiring nuclear weapons or supporting terrorism.

The two agreed to work on Afghanistan, where Obama is considering beefing up U.S. forces and where Moscow fears Taliban insurgency may spread Islamist militancy towards Russia.

Russia and the United States would cooperate on preparing a U.S.-sponsored conference in Afghanistan, Lavrov said.

The U.S. plan for a missile defence shield in Europe was touched on only broadly, a senior U.S. official told reporters.

Russia has said it sees the system as a threat, but Obama has assured them it is directed at missiles from Iran.

Clinton left after the meeting for Ankara for talks with Turkish officials.

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