Zurabishvili, 66, is running as an independent candidate for the largely ceremonial position, but has the backing of the ruling Georgian Dream party.
The election is seen as a test for the increasingly unpopular party, which is run by billionaire oligarch Bidzina Ivanishvili.
Zurabishvili took 39 percent of the first-round vote on 28 October, versus 38 percent for her rival, 60-year-old opposition leader Grigol Vashadze, who has the support of the party of exiled former president Mikheil Saakashvili.
A win for Vashadze would change the political landscape of the pro-Western, former Soviet state, and be a sign that Georgian Dream’s six-year dominance is ending.
Zurabishvili in Paris
Zurabishvili was born in Paris to a family that fled Georgia for political reasons in 1921.
She studied at the prestigious Paris Institute of Political Studies and at Columbia University in New York before leading a successful career at the French Ministry of Foreign Affairs, which culminated in an appointment as ambassador to Georgia in 2003.
In 2004, then-president Saakashvili granted her Georgian citizenship with the endorsement of French President Jacques Chirac, and she became Georgia’s foreign minister.
During her tenure, she negotiated the withdrawal of Russian military bases and established relationships that have facilitated Georgia’s aspirations to join Nato and the European Union.
After a confrontation with lawmakers led to her being sacked in late 2005, Zurabishvili set up an opposition party, Georgia’s Way, which she let until 2010.
She became an independent MP in 2016.
A previous bid to seek the presidency in 2013 was disqualified, due to her holding dual French and Georgian citizenship.
She announced in August that France had terminated her French citizenship at her request, so that she could submit her candidacy for the 2018 poll.
“The decision was not simple, but it was necessary,” Zurabishvili said at that time.
“The President of Georgia cannot simultaneously be a citizen of another country.”