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.
"I am very, very worried for Ukraine. First and foremost, this may lead to a long-term conflict that will completely destroy the health system," Lucica Ditiu, a Romanian physician and the Executive Director of the Stop TB Partnership in Geneva, was quoted as saying.
"I think we need to prepare for millions," she told reporters in Brussels, adding she was in favour of activating the EU's temporary protection directive to provide shelter for those people coming to the EU.
The conference is expected to strengthen the UAE’s global leadership position in climate action and sustainable development, in line with the vision and directives of its wise leadership.
The Mamiraua Institute, a research group of Brazil’s Ministry of Science, Technology and Innovation, said two more dead dolphins were found Monday in the region around Tefe Lake, which is key for mammals and fish in the area.
"Senator Doug Larsen, his wife Amy, and their two young children died in a plane crash last evening in Utah,” Hogue wrote in his email. "They were visiting family in Scottsdale and returning home. They stopped to refuel in Utah.”
Backed by the Sharjah Chamber of Commerce, the event played host to 500 specialised companies, unveiling the latest and most sought-after designs in watches, jewellery, and gold ornaments from around the world.