Ukraine had said earlier in the day that its forces were still holding out despite a Russian onslaught.
"Wagner units took control of the entire territory of Soledar. A cauldron has been formed in the centre of the city in which urban fighting is going on," Wagner head Yevgeny Prigozhin said in a statement cited by Russian agencies.
"The number of prisoners will be announced tomorrow," he added, giving no further details.
(Reporting by David Ljunggren; Editing by Sandra Maler)