Foreign minister Boerge Brende said his country does not have the capabilities to handle the request by the deadlines given so there was no point in continuing the discussions.
He said the US is looking at other alternatives but did not give details.
Mr Brende said Norway had not been able to identify a port that could receive the weapons and did not have the capacity to treat some of the waste products resulting from the destruction of the munitions.
In a webcast news conference, Mr Brende said the US agreed there was no point in continuing "the evaluation of Norway as a place for this destruction".
Norway earlier this week said it was one of the nations that had been asked to take part in the destruction of 50 metric tons of mixed chemicals in the form of mustard gas and 300-500 metric tons of materials needed to make nerve agents.
The US and Russia set a mid-2014 deadline for the destruction of Syria's arsenal, which Mr Brende said was too tight for Norway.