The nearly four-year civil war has pushed an average of 1,800 South Sudanese into neighbouring Uganda every day for the past year, many of them women and children fleeing "barbaric violence", according to the UN's refugee agency (UNHCR).
Another one million have sought refuge elsewhere in the region, but it is Uganda -- one of the world's poorest countries and the size of the United Kingdom -- that has borne the brunt of the crisis.
Many who arrive have harrowing stories, such as Jasmine Ramadan, a refugee now working as a shopkeeper in Uganda, who told AFP how her car was stopped by armed men who took her 13 -year-old daughter into the woods to rape her.
She pleaded with them to take her instead. They gave in and she ended up being raped by six men. Her daughter, hiding in the bush nearby, witnessed everything.
The conflict initially pitted Machar's ethnic Nuer against Kiir's Dinka, but since the collapse of a peace agreement in 2015, the war has engulfed other ethnic groups and local grievances.
Thousands of people have been killed by the violence, which plunged part of the country into famine earlier this year.
"The number of hungry and displaced South Sudanese is overwhelming," said International Committee of the Red Cross (ICRC) president Peter Maurer.
Another million South Sudanese have fled to Sudan, Ethiopia, Kenya, the Democratic Republic of Congo and the Central African Republic, according to UN numbers. Two million are internally displaced.
"Recent arrivals continue to speak of barbaric violence, with armed groups reportedly burning down houses with civilians inside, people being killed in front of family members, sexual assaults of women and girls, and kidnapping of boys for forced conscription," the UNCHR said in a statement.
UNHCR estimates it will need USD 674 million this year alone to help the refugees in Uganda, but said that so far it had received just a fifth of that amount.
Schooling has also been affected.
"Normal classrooms will have at least 53 or 55 students. But on average here, one classroom is having 110 pupils under a teacher, under the same roof," said Simon Abukua, principal of the Ocea Primary School.
While there is no denying the gravity of the situation, questions have been raised over the accuracy of the number of refugees really in Uganda.
Elizabeth Pfifer, Uganda country representative for the Catholic Relief Services (CRS) charity, says only the planned biometric registration of refugees will "help to relieve some of the inconsistencies and questions around the actual numbers."
Uganda has long been praised for having one of the world's most progressive refugee policies.
The government allows refugees to work and move around the country while communities in the north have donated land for settlements.
However the scale of the crisis has overwhelmed local communities and strained their hospitality.
The figure of one million refugees arriving in Uganda compares with a similar number of migrants who arrived in Europe during the height of its migrant crisis in 2015.
"If the government is changed and we have a new leader with a new government system, maybe I will come," said Ramadan.
"Because, what they did to me... I don't feel like going back.
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