In a gesture of solidarity, the council is also severing relations with a twin town in Russia.
The leader of the local authority has issued a statement responding to the situation in Eastern Europe and expressing the council’s support for the people of Ukraine.
Councillor Amanda Hopgood said: “We share the horror and sadness felt by people around the world at the situation in Ukraine and our thoughts are very much with those who live there, as well as their family members and friends in different countries.
“As a gesture of our strong solidarity with and support for Ukraine, we are very proud to be flying its national flag at County Hall.”
She added: “We are an inclusive and welcoming county and as part of that have developed a number of town twinning arrangements over the years, with communities in Germany, Slovakia, Denmark, France, Spain, America and Hungary.
“In 1968, County Durham was first linked with the Russian City of Kostroma although there have been no active projects in place through this twinning for some time.
“In light of the military action being carried out by Russia, and in line with steps already taken by other councils across the country, we will be ending the twinning arrangements with Kostroma as a show of our opposition.
"We wish to make it clear we do not blame people in Kostroma for the actions of their country’s leaders.
“Our thoughts remain firmly with Ukraine.”