Infectious disease experts expressed alarm about Ebola’s spillover into Uganda, even though that country has long anticipated that possibility and administered a promising new vaccine to thousands of health workers, as have the health authorities in Congo.

“The confirmation of new Ebola cases in Uganda is tragic but unfortunately not surprising,” Dr. Jeremy Farrar, director of the Wellcome Trust, a London-based health research institution, said in a statement. “This epidemic is in a truly frightening phase and shows no sign of stopping anytime soon.”

Some Ebola experts also expressed concern about the potential spread of Ebola to South Sudan, which like western Uganda also shares a porous border with the afflicted region of Congo. But unlike Uganda, South Sudan has been roiled by instability for years and is not considered well prepared to handle any possible outbreak.

“South Sudan is a whole different ballgame,” Adrian Ouvry, humanitarian adviser at Mercy Corps, an international charity, said by phone from South Sudan, where he was on a working visit. “It is much less developed than Uganda or even the D.R.C.”

The Congo outbreak is the second-deadliest on record, infecting more than 2,071 people and causing at least 1,396 deaths in the country as of June 10, according to the W.H.O. Its epicenter in a conflict zone has complicated efforts to contain the disease.

Health workers, including doctors, have been attacked and killed, and some treatment centers have been destroyed. In April, the Islamic State claimed its first assault in the affected area.

International health experts also have expressed worry about an acceleration in the number of Ebola infections. While it took about eight months to reach 1,000 cases, it has taken only a few months to surpass 2,000.