Nearly 1.37 million people have fled Ukraine since the start of the Russian invasion, according to the latest UN data Saturday.
The United Nations High Commissioner for Refugees (UNHCR) counted 1,368,864 refugees on its dedicated website at 1215 GMT, almost 160,000 more than the previous count on Friday.
Authorities and the UN expect the flow to intensify as the Russian army continues to advance into Ukraine, with fierce fighting still taking place around the capital Kyiv.
"1.3 million people have now fled Ukraine to seek safety. Unless there is an immediate end to the conflict, millions more are likely to be forced to flee," the UNHCR said in a tweet on Saturday.
According to the UN, four million people may seek to leave the country to escape the war.
Before the conflict, Ukraine had more than 37 million people in areas controlled by Kyiv – which does not include Russian-annexed Crimea or separatist-controlled areas.
Poland, which has championed the cause of Ukrainian refugees and where US Secretary of State Antony Blinken arrived on Saturday for talks with the country's top officials, is hosting by far the largest number of refugees arriving since the start of the Russian invasion.
In total, there were 756,303 refugees in Poland on Saturday, according to the UNHCR count, 106,400 more than on Friday and 55.3 percent of the total.
Before the crisis, Poland was already home to about 1.5 million Ukrainians, most of whom came to work in the EU member state.
Hungary has taken in 157,004 people, or 11.5 percent of the total, and almost 12,300 more than the previous day, according to the UNHCR.
The country has five border crossings with Ukraine and several border towns, including Zahony, have turned public buildings into relief centres, where Hungarian civilians offer food or assistance.
The UNHCR had not updated the number of refugees in Moldova on Saturday, which stood at 103,254 on Friday, or 7.5 percent of the total.
The head of the UN refugee agency, Filippo Grandi, was in Moldova on Friday, in the eastern town of Palanca where refugees are arriving.
"Today at Palanca I saw thousands and thousands of people streaming across Ukraine's border crossing with Moldova. Thousands of stories of separation, anguish, and loss. A difficult day, but much respect for the many dedicated Moldovan officials and people helping the refugees," Grandi said in a tweet on Friday evening.
According to the UNHCR, some of the refugees are continuing on to Romania or Hungary, often to reunite with family.
Some 101,529 people have fled Ukraine and taken refuge in Slovakia, 7.4 percent of the total, according to the UNHCR.
In Romania, the UNHCR counts 63,192 refugees, about 4.6 percent of the total.
Two camps have been set up, one in Sighetu Marmatiei and the other in Siret.
UNHCR also said that 133,876 people, around one in 10, had continued on to other European countries after crossing the Ukrainian border.
The number of people taking refuge in Russia remained unchanged at around 53,000, or 3.9 percent of the total.