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The death toll – which accounts for at least 36 fatalities a day since violence broke out in April – includes civilians, Ukrainian military personnel and rebel fighters.
More than 300 have been killed in the past week alone.
More than 6,000 people have been wounded since April the UN added in an updated report on Sunday.
There was a spike in casualties in the past three weeks when Ukrainian forces pressed ahead with an offensive to retake a number of vital cities in the east.
Rebel sources said that shelling had intensified in the cities of Donetsk and Luhansk, in particular, adding to the civilian death toll.
In the past week, the Ukrainian military appeared to consolidate its hold of Luhansk, and continued to tighten its noose around Donetsk, considered the main rebel center.
Nearly 600,000 civilians have left their homes in Donetsk as fighting and shelling intensifies.
UN High Commissioner for Human Rights Navi Pillay has told reporters that there is an urgent need to end the fighting in the east to spare more human casualties.
She wants to see the violence end before more civilians “are forced to flee, or face intolerable hardships trapped inside the conflict zones”.
“All those involved in the hostilities in the affected areas of the east must at all times comply with the principles of distinction, proportionality and precaution. This is particularly important in densely-populated areas,” she added.
Meanwhile, Russia and the Ukraine appear to have carried out the exchange of captured soldiers. Last week, Kiev showed on national television 10 Russian paratroopers it said had crossed into Ukraine to aid pro-Moscow rebels.
Russia had insisted its soldiers had crossed the border by accident and were lost.
On Friday, as pro-Moscow rebels carried out a series of operations which cut short Kiev’s offensive, 63 Ukrainian soldiers retreated by entering Russian territory.
Kiev and its Western allies have accused Moscow of directly intervening to prevent a defeat of the rebels.