Place a saucer with two metal spoons in a flat place in your freezer. Rinse rosemary well under cold water, pat dry between two clean towels, and set aside.
In a 16-quart copper preserving pan or a stainless steel kettle, combine berries with sugar and lemon juice. Place pan over medium-low heat and cook, stirring frequently. After a few minutes, as the juice starts to run and the mixture begins foaming a little around the edges, gradually raise heat to high, stirring often.
Boil hard for approximately 20-30 minutes, gently scraping the bottom every few minutes with a heatproof rubber spatula to be sure mixture is not sticking. If it begins to stick, reduce heat slightly, making sure it continues at a rapid boil. Continue to cook, scraping bottom frequently, until foam subsides; the mixture acquires a darker, shinier look; and the berries appear softened and saturated with liquid, approximately 25 minutes total.
Turn off heat. Do not stir. Let mixture sit a moment, then use a metal soup spoon to carefully scrape all the white foram off the top of the mixture. When you have removed every last bit of white, stir in the Marsala, little by little, tasting as you go. The flavor should be present, but not overpowering. Return to medium heat and cook, stirring frequently. If necessary, gradually lower the heat to prevent scorching.
After 3-5 minutes, the jam should again look glassy and dark. To test for doneness, remove the jam from heat and take a small representative half-spoonful (one containing both the liquid and the more solid portions of the mixture) and carefully transfer it onto one of your frozen spoons. Replace the cold spoon in the freezer for 1 to 2 minutes.
Remove from freezer and nudge it gently with your finger. It should by this time be neither warm nor cold; if still warm, put it back in the freezer for a moment. Tilt the spoon vertically to see how quickly your jam runs; if it runs slowly, and if it has thickened to a gloppy consistency, it is done. If it runs very quickly, or appears watery, cook for another minute or two, stirring, and test again, repeating more times if necessary. This jam, while spreadable, has a relatively loose texture.
Turn off heat but do not stir. Skim all remaining foam from the surface of your jam, then stir well to be sure berries and liquid are evenly distributed. Place rosemary stalks into jam, stir, leave 1 minute, taste (carefully- the texture will be very hot), and either remove sprigs or leave to steep another moment.
Pour into jars sterilized according to manufacturer's instructions and store in the refrigerator for up to 4 weeks.