Apple Juice with Blackberry Syrup

To make syrup: In a small saucepan, combine berries, sugar, and 1/2 cup plus 2 tablespoons of water. Bring to a simmer over medium heat and cook gently for 15 minutes, stirring occasionally. Remove from heat and pass through a sieve, pressing with the back of a wooden spoon so that only the seeds and skin are left behind. Discard contents of sieve. Stir lemon juice into syrup. You can use it immediately, or let cool and refrigerate up to 3 days. To serve: Heat apple juice (or simply use water, if preferred) in a pot or microwave. Stir about 2 tablespoons hot blackberry syrup into each cup of juice, until dissolved. Garnish with apple slices or blackberries, if desired.