Firstly place 5 saucers into the freezer and then sterilize the jars and lids by placing them in an oven heated to 100°C for 20 minutes.
Roughly chop the apples without peeling or coring, then place in a large preserving pan with the blackberries. Heat gently until all the fruit has softened.
Remove from the heat, then pass everything through a sieve or food mill.
Replace the seedless fruit pulp back into the saucepan and keep the seedy fruit pulp for another purpose (like blackberry gin or vinegar).
Add the lime juice and sugar to the saucepan and heat gently so all the sugar has dissolved. Once dissolved, bring to a rolling boil. Boil for 5 minutes then turn off the heat and place on one of the cold saucers from the freezer. Leave for 1 minute then push the jam with your finger. If the jam wrinkles on the surface it’s ready. If not, then turn the jam back on a boil for a further 3 minutes. Turn off the heat and try the saucer test again.
Once the jam is ready remove the scum from the surface of the jam and then stir in the lime zest.
Decant the jam into the sterilised jars, screw the lids on tightly and store until ready to use.