A production of 42.9m hectolitres (hl) (944m gallons) is forecast for 2016, down from last year's 47.8m hl.
The harvest is expected to be down by about a third in the Champagne region.
Prices will rise, although one French wine trader said the Brexit effect would be a greater inflationary pressure for UK buyers.
Jean-Pierre Hourlier, who sells mid-market French wines in the UK, said: "Prices are going up across the board because of the euro and the impact on transport because of Brexit."
After the British vote on 23 June to leave the EU, the pound slumped against the euro, falling from €1.3 in late May to €1.17 now. Imports priced in euros are therefore more expensive.
Mr Hourlier's father, Pierre, told the BBC that the heatwave in southern France - hitting the big wine regions of Bordeaux and Languedoc - was a particular problem now.
"When the vineyards run out of reserves the supermarkets suffer more than niche wine merchants like us," he said.
But he added: "Even if it's a difficult vintage they'll still produce good wines - there will be quality, just a reduced amount."
He said French wine producers had had some "dreadful" times in the 1990s too, with low quantities and poorer quality.
"It's too early to dramatise - when the quantities are in, then we'll know," he added.
The figures from the agriculture ministry's Agreste data service showed the 2016 harvest 7% down on the 2011-2015 average of 46.1m hl.