Rinse the fish bones well in several changes of cold water. Place them in a medium stockpot. Cook, covered, over low heat, until their juices are released, about 10 minutes. Stir frequently to avoid sticking.
Add enough cold water to cover, and the bouquet garni. Bring slowly to a boil, skimming surface until no trace of scum remains. Reduce heat to simmer, and cook for 25 minutes.
Strain through a fine mesh sieve lined with damp cheesecloth. Cool. Store in an airtight container, refrigerated, up to 3 days or in the freezer for up to 1 month.