It is the first binding public vote on the treaty, which must be ratified by all 25 member states to become law.
The French decision is expected to be close on an issue that has polarised the nation.
Correspondents say a French “Yes” could give the pact new momentum but a “No” may trigger more opposition and threaten the EU’s first constitution.
The outcome in France is importance because it is traditionally a leader in the continent and a key power in the EU-organisation.
Nine countries have so far ratified the constitution, which aims to streamline EU decision-making.