A 50-year-old woman infected with Ebola has died in western Uganda, a day after her five-year-old grandson died.
They are the first cases reported there since the virus spread over the border from Democratic Republic of Congo, where nearly 1,400 people have died.
Three suspected Ebola patients have also run away from an isolation centre in south-western Uganda.
The head of a major medical charity has called the latest outbreak of Ebola in central Africa "truly frightening".
Dr Jeremy Farrar, the director of the Wellcome Trust, said the epidemic was the worst since that of 2013-16 and has showed "no sign of stopping".
He said the spread was "tragic but unfortunately not surprising". He warned that more cases were expected, and a "full" national and international response would be needed to protect lives.
On Friday the World Health Organization (WHO) will decide whether the outbreak should now be deemed a global health emergency.
The grandmother had lived in DR Congo, where her husband recently died of Ebola. Her daughter had travelled from Uganda, where she lives with her Ugandan husband, to the Congolese town of Beni with her children to help care for him.
After the grandfather died they all travelled on Sunday to Uganda's Kasese district, where the five-year-old son and his grandmother then subsequently became sick and later died.
The three-year-old son is now confirmed to have Ebola. He, his parents and two of his siblings have been repatriated at their request to DR Congo, Uganda's health ministry says.
Twenty-seven people are said to have been in contact with the three confirmed cases in Uganda. They have been restricted to their homes and will be vaccinated against Ebola.
The people who fled from a hospital isolation unit had been found to have high temperatures when they crossed the border from DR Congo to the Ugandan district of Kanungu, which is about 150km (93 miles) south of Kasese. Medical workers did not get a chance to take samples of their blood to send for testing before their escape.
How prepared is Uganda?
Analysis by Patricia Oyella, BBC Africa, Kampala
This isn't the first time the virus has struck the country. Outbreaks in 2012, 2007 and 2000 have prompted the health ministry to build capacity, says Director of Clinical Services Dr Charles Olaro.
Public health information campaigns broadcast to at-risk areas have been key in preventing the spread since the first case of Ebola was reported over the border in DR Congo last August. Mass gatherings, including market days and prayers, have been cancelled. Market days in the town of Kasese attract an estimated 20,000 people at the border area.
Uganda's health ministry and the WHO said a rapid response team had been dispatched to identify others at risk. The country has already vaccinated about 4,700 health workers against the disease, according to a joint statement by WHO and Ugandan health officials. Ebola screening centres are in operation along the border with DR Congo as well as other major entry and exit points.
Authorities have identified 22 high-risk districts - places that are close to the border and have high levels of movement - and deployed medical monitors to spot and manage any cases they find.
A database of experts is on hand to deal with different scenarios, the health ministry says, and Uganda also benefits from the expertise of health workers it deployed to contain previous outbreaks in West Africa.
What's the situation in DR Congo?
Nearly 1,400 people have died over the last 10 months - around 70% of all those infected.
The outbreak is the second-largest in the history of the disease, with a significant spike in new cases in recent weeks.
Only once before has an outbreak continued to grow more than eight months after it began - that was the epidemic in West Africa between 2013-16, which killed 11,310 people.
Some have predicted it could take up to two more years to bring to an end. The WHO has twice ruled that this Ebola outbreak is not yet a global emergency. I
Efforts to contain the spread have been hindered by militia group violence and by suspicion towards foreign medical assistance.
Nearly 200 health facilities have been attacked in DR Congo this year, forcing health workers to suspend or delay vaccinations and treatments. In February, medical charity Médecins Sans Frontières (MSF) put its activities on hold in Butembo and Katwa - two eastern cities in the outbreak's epicentre.
What is Ebola?
- Ebola is a virus that initially causes sudden fever, intense weakness, muscle pain and a sore throat.
- It progresses to vomiting, diarrhoea and both internal and external bleeding.
- People are infected when they have direct contact through broken skin, or the mouth and nose, with the blood, vomit, faeces or bodily fluids of someone with Ebola.
- Patients tend to die from dehydration and multiple organ failure.