Refugees from Ukraine are pictured after crossing the Ukrainian-Polish border in Korczowa on Thursday. AFP
Hungary, Moldova, Slovakia and Romania have also welcomed tens of thousands of refugees. Countless people have also been displaced inside Ukraine.
Grandi hailed the "remarkable" response of governments and local communities in surrounding countries who have accommodated the million refugees.
The UN children's agency Unicef said that half of those who had left Ukraine were youngsters.
"Half a million children have already fled Ukraine to neighbouring countries, with the number of refugees continuing to grow," it said in a statement.
Unicef said it was setting up "safe spaces" for children and mothers to access services along transit routes.
Treat everyone equally: UN leaders
UN leaders also pleaded for everyone fleeing to be treated equally, voicing alarm at reports of African and Asian nationals facing discrimination at the border.
"In just seven days we have witnessed the exodus of one million refugees from Ukraine to neighbouring countries," UN refugee chief Filippo Grandi tweeted.
"Unless there is an immediate end to the conflict, millions more are likely to be forced to flee Ukraine," he warned.
According to data from Grandi's UNHCR agency, 1,045,459 people have now fled Ukraine since Russian President Vladimir Putin launched the full-scale invasion on Feb. 24.
Grandi said that in almost 40 years working in refugee emergencies, "rarely have I seen an exodus as rapid as this one."
More than half of those who have fled Ukraine — nearly 548,000 people — have crossed west into Poland, UNHCR said.
Poland's border guards said more than 600,000 people in total had crossed the frontier — many will have since moved on elsewhere in Europe — with 56,400 people alone crossing on Thursday, by 1400 GMT.
The UN's aid chief Martin Griffiths said the number of refugees was "going to keep mounting very quickly, by the hour."
With Russian forces closing in on major cities, people's risk calculation may shift from staying to leaving and "we will see more displacement", he told AFP.
He warned the parties to the conflict that they were responsible for protecting civilians within Ukraine and also their "safe outflow, if they so desire".
UN human rights chief Michelle Bachelet said the million refugees had often spent days travelling by bicycle or on foot in freezing conditions. But she raised concerns for fleeing non-Ukrainians who, after heading for the borders, faced discrimination.
"I commend the welcome that Ukrainians leaving the country have received. This welcome must be extended to all those fleeing conflict, regardless of their citizenship, ethnicity, migration or other status," she told the UN Human Rights Council.
"There have been disturbing indications of discrimination against African and Asian nationals while fleeing."
The International Organisation for Migration's leader Antonio Vitorino also voiced strong concerns, saying fleeing foreign workers and students were facing "heightened risk and suffering." "I am alarmed about verified, credible reports of discrimination, violence and xenophobia against third-country nationals attempting to flee the conflict in Ukraine," the IOM chief said in a statement.
"Discrimination on the basis of race, ethnicity, nationality or migration status is unacceptable. I deplore any such acts and call on states to investigate this issue and address it immediately."
The Human Rights Council's African nations were also disturbed by reports of Africans "being denied the right to cross the border to safety."
"Everyone has the right to cross international borders during a conflict," Ivory Coast's representative said, on behalf of the group. "Show the same empathy and support for all people fleeing war, regardless of their racial identity," he urged.Agence France-Presse
Ukraine dismissed Moscow's offer to set up humanitarian corridors from several bombarded cities on Monday after it emerged some routes would lead refugees into Russia or Belarus.
Shabia Mantoo, a UNHCR spokeswoman, said the latest and still growing count had 281,000 in Poland, more than 84,500 in Hungary, about 36,400 in Moldova, over 32,500 in Romania and about 30,000 in Slovakia.
"To the world: what is the point of saying 'never again' for 80 years, if the world stays silent when a bomb drops on the same site of Babi Yar," Ukrainian leader Volodymyr Zelensky asked in a tweet.
The typing centres affirmed that the entry permit application was activated since last Friday and covered all types of residence visas, noting that the holder of a UAE residence visa who stayed abroad for more than 6 months can apply for the entry permit.
Jabal Jais in Ras Al Khaimah recorded the lowest temperature of 1.9°C at 4:15 am on Sunday. Jabal Al Rahba in RAK also recorded 4.2°C. and Jabal Mabrah recorded 4.6 °C, while Rukna in the city of Al Ain in Abu Dhabi, recorded 6.7°C.
The court indicated that he based his claim on a copy of the medical report, which was not sufficient to prove that the injury occurred as a result of work.