"Should the regime fail to act, there will be consequences," Kerry told the Security Council after the vote sealing a US-Russian agreement.
Kerry hailed the deal, reached after prolonged negotiations, as a breakthrough.
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"Provided this resolution is fully implemented, we will have eliminated one of the largest chemical weapons programs on Earth from one of the most volatile places on Earth."
But Kerry said that chemical weapons were only one issue and that a solution was needed to end the 30-month civil war that has killed more than 100,000 people.
The world "still carries the burden of doing what we must to end mass killing by other means," Kerry said.
"We must work together with the same determination, the same cooperation that has brought us here tonight, in order to end the conflict that continues to tear Syria apart even this very day."
The resolution, approved unanimously by the 15-member Council, orders Syrian President Bashar al-Assad to destroy chemical weapons in the wake of an August 21 attack that US intelligence says killed 1,400 people.
In a compromise, the Security Council agreed to state that it would return and impose measures under Chapter VII -- which makes a resolution binding -- if Syria does not comply.
Russia, which proposed the weapons agreement to prevent US and French military action in Syria, would be likely to fight any future move to order war or sanctions against its ally Assad.
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