Army spokesman Philip Aguer said soldiers entered the town, capital of Jonglei State and situated 200 kilometres (130 miles) north of the capital Juba, in the afternoon following days of fierce fighting.
Uganda's army spokesman Paddy Ankunda also confirmed that Ugandan troops, who have been backing South Sudan's President Salva Kiir, took part in the offensive.
"Today the gallant SPLA (Sudan People's Liberation Army) forces entered Bor, they have defeated more than 15,000 forces of (rebel leader) Riek Machar and frustrated his plans to attack Juba and install himself as the ruler of South Sudan," Aguer told reporters.
Aguer said the battle had left "many dead", but did not give figures. Rebel military spokesman Lul Ruai Koang confirmed the town had fallen, but insisted that the rebels chose to make a "tactical withdrawal" to reorganise.
"It is not a big issue. There are 11 counties in Jonglei state we are in control of nine counties. So if we are in control of nine counties, why should we waste our time on just one small county without even a population? There is no population in Bor, the entire population fled," he said.
"If there are people captured they should be kept in regards to the international law," the spokesman said.
Heavy fighting was reported to be continuing in and around the key oil town of Malakal, capital of Unity State and one of the main battlefields since fighting erupted last month between rival forces loyal to President Kiir and his sacked deputy Machar.
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