"We are doing so again as you work to overcome an invasion of your sovereign territory and an illegitimate unilateral attempt to change your country's borders by force that has been universally condemned by the free world.
"Russia's actions have cast grave doubt on Russia's intentions and on its reliability as an international partner - not just in Georgia but across this region and, indeed, throughout the international system."
Before leaving for Ukraine, Mr Cheney visited a military airport near Tbilisi to meet US workers and military personnel who are carrying humanitarian aid to the country.
In Kiev, the vice-president is expected to hold talks with President Viktor Yushchenko and Prime Minister Yulia Tymoshenko.
The alliance between the two leaders has faltered in recent weeks, with Mr Yushchenko being much more critical of Russia than Ms Tymoshenko.
Georgia has been a significant troop contributor to US operations in Iraq, and it is a key link in the only energy export route from Central Asia westwards that does not pass through Russian territory.
However, the aid package announced on Wednesday is limited to helping resettle refugees and rebuild Georgia's infrastructure, and the BBC's diplomatic correspondent Jonathan Marcus says it remains unclear how far the US and its Nato allies are prepared to go in re-arming its military.
On Wednesday, the International Monetary Fund (IMF) also announced that Georgia was to receive a $750m (£422m) loan.
Russian President Dmitry Medvedev has accused the US of helping Tbilisi build up its military, and criticised its decision to send humanitarian aid to Georgia aboard military ships.
Faced with a chorus of international calls for Russia's isolation as a result of the war, Mr Medvedev said Moscow did not fear being expelled from the G8 group of rich nations, nor did it fear Nato cutting ties with his country.
Early this week, EU leaders agreed to suspend talks on a new partnership agreement with Moscow until Russian troops had withdrawn from Georgia, but they did not threaten sanctions.
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