"The fighting is very heavy. There is fighting on the outskirts of the town. It's a very big, coordinated attack," an independent source said.
The fighting appeared the heaviest to take place since the government of President Salva Kiir and rebels loyal to former vice president Riek Machar signed a ceasefire agreement in neighbouring Ethiopia on January 23.
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Malakal, situated on the bank of the White Nile, is one of three state capitals that were in rebel hands but which were recaptured by South Sudan's government, backed by Ugandan troops, before the ceasefire was signed.
The conflict in South Sudan, the world's youngest nation which won independence from Khartoum less than three years ago, erupted in the capital Juba on December 15 but quickly spread across the country.
The fighting in South Sudan has left thousands dead and displaced close to 900,000 people.