In a Facebook posting on January 3, the 128th Mountain Brigade said the soldier in the video belongs to the unit.
The Ukrainian military unit said statements by the soldier in the video were made under duress and that all efforts were being taken to secure his release.
Few other confirmed details of the fate of the soldier, identified as Andriy Kachynskyy, were immediately available.
Separatist-controlled media said he was seized while trying to enter a separatist-controlled area of the Donetsk region on December 29.
News of the captured Ukrainian soldier comes amid a new truce in eastern Ukraine.
The cease-fire took effect on December 29 and is scheduled to last until January 7.
Ukrainian government forces have been fighting Russia-backed separatists in eastern Ukraine since April 2014, shortly after Russia seized Ukraine's Crimean Peninsula and forcibly annexed it.
Some 10,300 people have been killed in the fighting since early 2014.
