Add the birch syrup to the bottom of a quart mason jar.  Bring the water to a boil and pour it over the birch syrup.  Stir to dissolve.

Pitch 1/4 packet of wine yeast into a few tablespoons of water and allow the yeast to dissolve.

Allow the birch syrup mixture to come to room temperature, and then pitch the yeast (add it in).  

Cap with a mason jar fermentation kit, and allow the mixture to ferment at room temperature for a few weeks until visible fermentation has stopped.  If you're using white sugar and birch sap, it may be done in less than a week.

Carefully pour the birch beer off into another container, leaving any yeasty sediment behind in the mason jar.  Bottle in a simple flip-top Grolsch bottle and allow it to age for at least a few days, but preferably 2 weeks, before drinking.