The venerable distillery, which was bought by LVMH luxury goods group in 1987, sold 83.8 million bottles of drink last year.
Half of its output went to the US and a quarter to Asia, excluding Japan.
The new bottling and logistics facility in the village of Salles d'Angles -- a few kilometres from the town of Cognac after which Hennessy's variety of brandy is named -- will give a much-needed boost to production, LVMH CEO Bernard Arnault said.
"It's not a bad problem to have but our stocks are very low and we cannot meet demand. We're short on bottles and on product because of the success of our products across the Atlantic and in China," Arnault said at Wednesday's launch.
Hennessy aims particularly to turn out more bottles of the VS (Very Special) cognacs popular in the US and the VSOP (Very Special Old Pale, aged for at least four years) and XO (Extra Old, aged for over 10 years) varieties beloved of big-spending Chinese customers.
Hennessy CEO Bernard Peillon told AFP he aimed to quadruple output over the next 25 years.