Two other deaths were reported in a government-controlled area.
Russian news agencies cited rebel armed forces chief Vladimir Kononov as saying the dead included fighters and civilians. The figure could not immediately be verified.
The Ukrainian government and rebels have accused one another of sparking the latest round of unrest, which marks some of the worst violence since a cease-fire deal was reached in February.
"The Ukrainian side has carried out a provocation and started shelling our positions all along the front," the Interfax news agency quoted Kononov as saying.
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"The enemy is trying to advance. Ukrainian troops are repelling all attacks and successfully holding onto their positions," Lysenko said.
Russia's Tass news agency quoted the chief physician at the trauma centre in Donetsk, Alexander Oprishchenko, as saying his hospital had admitted 60 wounded people for treatment today, of whome five were in a grave condition.
Implementation of the cease-fire has foundered as low-intensity battles raged along several points of the 450-kilometer ront. Negotiators from the warring sides met for talks in Belarus yesterday, but the session concluded without obvious progress.