Former spy Sergei Skripal and his daughter were poisoned by a military-grade nerve agent of a type developed by Russia, Theresa May has told MPs.
The PM said it was "highly likely" Russia was responsible for the Salisbury attack.
The Foreign Office summoned Russia's ambassador to provide an explanation.
Mrs May said if there is no "credible response" by the end of Tuesday, the UK would conclude there has been an "unlawful use of force" by Moscow.
The chemical used in the attack, the PM said, has been identified as one of a group of nerve agents known as Novichok.
Mrs May said: "Either this was a direct action by the Russian state against our country, or the Russian government lost control of its potentially catastrophically damaging nerve agent and allowed it to get into the hands of others."
She said Foreign Secretary Boris Johnson had told the ambassador Moscow must provide "full and complete disclosure" of the Novichok programme to international body the Organisation for the Prohibition of Chemical Weapons.
Mrs May said the UK must stand ready to take much more extensive measures, and these would be set out in the Commons on Wednesday should there be no adequate explanation from Russia.
Retired military intelligence officer Mr Skripal, 66, and his daughter, Yulia, 33, were found slumped on a bench in Salisbury city centre on Sunday 4 March. They remain in a critical but stable condition in hospital.
Det Sgt Nick Bailey, who fell ill attending the pair, remains seriously ill, but has been talking to his family.
Mr Skripal was convicted by the Russian government of passing secrets to MI6 in 2004, but given refuge in the UK in 2010 as part of a "spy swap".
Home Secretary Amber Rudd will chair a meeting of the government emergency committee Cobra on Tuesday to discuss the latest developments in the case.
What are Novichok agents?
The name means "newcomer" in Russian, and applies to a group of advanced nerve agents developed in secret by the Soviet Union in the 1970s and 1980s
One chemical - called A-230 - is reportedly five to eight times more toxic than VX nerve agent, which can kill a person within minutes
Some are liquids, others are thought to exist in solid form. Some are reported to be "binary weapons", meaning they are typically stored as two less toxic chemicals which when mixed, react to produce the more toxic agent
One variant was reportedly approved for use by the Russian military as a chemical weapon
Designed to escape detection by international inspectors, their existence was revealed by defectors