Syria has said it wants the weapons destroyed outside the country, which is in the throes of civil war.
Albania had been considered the strongest hope, and few diplomats expected the NATO country of 2.8 million people to reject what Prime Minister Edi Rama said had been a direct request from the U.S.
But the plan was unpopular in Albania, and young protesters had camped outside Rama's office to oppose it, fearing it would be a health and environmental hazard.
Chemical weapons have to be incinerated at extremely high temperatures or neutralized using other chemicals both costly, risky and time-consuming operations that require specialized machinery.
In a televised address from the capital of Tirana, Rama said that it was "impossible for Albania to take part in this operation" an announcement that brought a loud cheer from some of the 2,000 protesters.
Rama said he rejected the request because other countries, which he did not identify, were not prepared to be a part of the operation.
The OPCW's Ellahi said: "It was a sovereign decision that Albania has taken."
In Washington, State Department spokeswoman Jan Psaki said the decision would not hurt U.S.-Albanian relations.
"We appreciate Albania looking seriously at hosting the destruction of chemical weapons," she said. "The international community continues to discuss the most effective and expeditious means for eliminating Syria's chemical weapons program in the safest manner possible."
Albania is one of only three nations that have declared a chemical weapons stockpile to the OPCW and destroyed it. The U.S. and Russia have also declared stockpiles but have not yet completed their destruction.
Tirana has been an avid supporter of Washington since the U.S. and NATO intervened with airstrikes in 1999 to stop a crackdown by Serb forces on rebel ethnic Albanians in neighboring Kosovo.
"Without the United States, Albanians would never have been free and independent in two countries that they are today," Rama said in an apologetic speech.
But the relationship was not enough to convince the hundreds of protesters.
"We don't have the infrastructure here to deal with the chemical weapons. We can't deal with our own stuff, let alone Syrian weapons," said 19-year-old architecture student Maria Pesha, among the protesters camped out overnight outside Rama's office. "We have no duty to obey anyone on this, NATO or the U.S."
Albania has had problems with ammunition storage in the past.
In 2008, an explosion at an ammunition dump at Gerdec near Tirana killed 26 people, wounded 300 others and destroyed or damaged 5,500 houses. Investigators said it was caused by a burning cigarette in a factory where some 1,400 tons of explosives, mostly obsolete artillery shells, were stored for disposal.
Wherever it happens, the destruction of Syria's weapons will be overseen by experts from the OPCW, which won the Nobel Peace Prize this year for its efforts to eradicate poison gas around the world.
Just getting Syria's weapons out of the war-torn country will be a major challenge.
Sigrid Kaag, the Dutch diplomat running the joint U.N.-OPCW mission in Syria, said her team is working "in an active war zone, in an extreme security situation with serious implications for the safety" of all personnel.
Norway has offered a cargo ship and naval frigate to help transport the chemicals.
The disarmament operation started more than a month ago with inspections. Machinery used to mix chemicals and fill empty munitions was smashed, ending the Syrian government's capability to make new weapons.
The disarmament mission stems from a deadly Aug. 21 attack on rebel-held suburbs of Damascus in which the United Nations determined that sarin was used. Hundreds of people were killed. The U.S. and Western allies accuse Syria's government of responsibility, while Damascus blames the rebels.
Syria's conflict, now in its third year, has killed more than 120,000 people, according to activists. It started as an uprising against President Bashar Assad's rule but later turned into a civil war.
The Britain-based Syrian Observatory for Human Rights, which relies on activists on the ground, said Friday that a government airstrike the previous night in northern Syria killed a senior rebel figure and wounded two commanders and the spokesman of the Tawhid Brigade, the main rebel outfit in Aleppo province.