Fine Marne" most likely refers to Fine de la Marne, a French brandy or marc from the Champagne region, made from grapes from the Marne Valley. It is a type of aged brandy or marc de champagne, similar to Italian grappa, known for its complex flavors derived from oak barrels. The term could also be a more general reference to the Marne river, a tributary of the Seine. 
Fine de la Marne brandy
What it is: A type of aged brandy or marc, with "Fine de la Marne" referring specifically to brandy made from grapes from the Marne Valley region of Champagne.
Production: It is produced from distilling the fermented juice or pressing the skins and seeds of grapes from the Champagne vineyards.
Aging: It is aged in oak barrels for several years, which imparts a deep amber color and complex flavors.
Flavor Profile: It is often described as smooth, full-bodied, and rich with notes of dried fruits, nuts, spices, vanilla, and caramel.