© APUS secretary of State John Kerry is greeted by Russian foreign minister Sergey Lavrov at the Russian ambassador's residence in Paris on Sunday
After four hours of talks in Paris, Mr Kerry said: "We will not accept any path forward where the legitimate government is not at the table."
Mr Kerry met Sergei Lavrov, his Russian counterpart, for talks that could shape Ukraine's future. The meeting came as Russian troops massed on the country's border.
The secretary of state said the US still considered Russia's actions in Ukraine, like the annexing of the Crimea, illegal and illegitimate. He said their discussions had touched on the rights of minorities, free and fair elections and the demobilisation of irregular troops.
Mr Lavrov said Ukraine could not function as a unified state, and should be a federation.
The talks followed a late-night phone call from Russian President Vladimir Putin to US President Barack Obama on Friday, when the two discussed potential diplomatic solutions to the Ukraine crisis. In a sign of the apparent urgency that both sides placed on the talks, Mr Kerry turned his aircraft round after a refuelling stop at Ireland's Shannon airport on Saturday to return to Paris for the meeting.
Russia restated before the talks demands it made two weeks ago as it moved to annex Ukraine's Black Sea peninsula of Crimea. They included military neutrality for Ukraine, a federal structure for the country, and promotion of Russian to an official state language alongside Ukrainian.
Moscow has stationed tens of thousands of extra troops and tank units on its border with Ukraine, fuelling fears that it may try to invade or push further into the country.
Ukraine's government expressed "outrage" late on Sunday at the proposals made by Russia's foreign minister.
"Russia's proposals for federalisation, a second official language, and referendums are viewed in Ukraine as nothing less than proof of Russia's aggression," Ukraine's foreign ministry said. "We sincerely regret that Minister S. Lavrov had to voice them."
It went on to describe Mr Lavrov's remarks as an "ultimatum", and said Russia "demands only one thing - the complete capitulation of Ukraine, its dismemberment, and the destruction of Ukrainian statehood".
After Mr Obama and Mr Putin's phone call on Friday, the White House released a statement stressing the possibility of a US-Russian compromise in the event that Moscow "pulls back its troops and does not take any steps to further violate Ukraine's territorial integrity and sovereignty". Washington has been pressing for international monitors to be deployed in Ukraine, with a mission from the Organisation for Security and Co-operation in Europe under way, and for early elections.
The Kremlin's statement on the call, however, struck a different note. It said Mr Putin had repeated his warnings of "extremists" in Ukraine who, he said, were "committing acts of intimidation towards peaceful residents" and certain government agencies "with impunity".
Moscow also called for a "fair and comprehensive settlement" to conflict in Transnistria, the pro-Russia breakaway Moldovan republic, which has been seen as another possible target for Russian military intervention.
Diplomats with knowledge of talks to date between Mr Kerry and Mr Lavrov said the Russian foreign minister appeared to have been given almost no negotiating flexibility over Russia's demands.
In an interview on Russian television at the weekend, Mr Lavrov denied that Russia had any plans to invade eastern Ukraine. In Donetsk, which has seen weekly anti-Kiev rallies with demonstrators sometimes numbering in the thousands, the situation was much calmer on Sunday, with only 300 people showing up to a planned pro-Russia rally.
However, Mr Lavrov reiterated Moscow's demands regarding federalisation and the promotion of Russian to an official state language, which he said Kiev had rejected. Andriy Deshchytsia, Ukraine's interim foreign minister, "said that our proposal was unacceptable because federalisation goes against the fundamental principles of Ukraine's state set-up", Mr Lavrov said.
"It is not clear why. I don't know anything about such principles. Second, the idea of making Russian Ukraine's second state language was also rejected as unacceptable.
"We can conclude that if the Ukrainian leadership persists in rejecting the idea of federalisation . . . if they continue to ignore ethnic Russians and the Russian language, the constitutional reforms that they have begun will not yield any sensible results," Mr Lavrov said.
Concerns increased of more tensions in Crimea after Crimean Tatars voted at a meeting in Bakhchisaray, their historic capital, to seek self-determination and appealed to international bodies to uphold their rights as a people.
At an extraordinary meeting of the Kurultay, the Tatars' national council, delegates adopted a resolution vowing to "launch political and legal procedures [for the] ethnic and territorial autonomy of the Crimean Tatar people on their historical territory, Crimea".
The measure appealed to the UN, Council of Europe, OSCE and other institutions to support them in their push for autonomy.
Russia's annexation of Crimea has worried the Tatars, who make up about 300,000 (15%) of the Ukrainian peninsula's 2 million population, and are uncertain whether to accept Russian citizenship and acquiesce to the takeover or remain in their homeland with "foreign" passports.
Many boycotted the March 16 referendum which was used to endorse Russia's widely condemned takeover of Crimea.
Additional reporting by Richard McGregor in Washington
The Russian Demands
● Military/political neutrality for Ukraine, to be guaranteed by Russia, the EU, US and a UN Security Council resolution.
● Creation of a constitutional assembly, with equal representation of Ukraine's regions, to carry out constitutional reform leading to a federal structure.
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● Russian as a second state language.
● After constitutional reforms, elections to all state and regional bodies.
● The confiscation of illegal arms and the clearance of buildings and streets occupied by protesters in Kiev.
● Respect the result of Crimea's referendum.
Comment: The hysteria surrounding the Russian request for Russian to be the second language in Ukraine smacks of hypocrisy. In the US, the Latino population speak Spanish, in Quebec, a significant part of the population speak French, and in countries like Australia, immigrant populations speak their own languages without fear or favour. Yet we are asked to believe that because Russians in Ukraine want to the right to speak Russian, this is supposedly proof of Russia's aggression. Doesn't add up really, does it?