While fresh blackberries, or even boysenberries, will give you the best results, this can be made with frozen berries if in a JAM (pun intended). Times are estimates. High altitude instructions are included.
Combine the berries and lemon juice in a heavy 4-qt saucepan. Bring to a boil over medium-high heat, stirring often.
Stir in the sugar, lemon zest, and cinnamon, and allow the mixture to boil to dissolve the sugar.
Reduce the heat to medium. Boil gently for about 15 minutes, stirring often, until the temperature on a candy thermometer reaches 220F (substract 2F degrees for each 1,000 feet above sea level).
Remove pot from the heat. With a long-handled spoon, stir the jam and skim off any foam that rises to the top.
Ladle the jam into hot, sterilized half-pint canning jars, leaving a 1/4-inch space on top. Place a piece of cinnamon stick in the jam in each jar. Seal the lid according to the manufacturer's instructions.
Cool the jars upright on a wire rack. Store in the refrigerator for up to 1 month.
For longer storage, process the jars in a boiling-water bath for 5 minutes after sealing. Add an additional minute of processing time for each 1,000 feet above sea level.
Cool on a rack and label. Store the jars in a cool, dry, dark place.