Bring the sugar and water to the boil in a small saucepan. Allow to simmer for one minute (swirl the pan to make sure the sugar has melted but do not stir). Once you have a clear, shiny, thick syrup, remove from the heat. Drop in the basil leaves, then decant the syrup into a glass jug or bowl to cool. Cover.
Cut the peaches into chunks and macerate them with the lemon zest and the lemon juice. (I just peel this in strips with a swivel peeler here as everything will be blitzed and strained eventually but you can finely zest it if you prefer.)
When the syrup has cooled, pour it over the peaches and leave the whole lot to infuse for a few hours at room temperature (or covered in the fridge overnight).
Blend everything (leaves, zest and peach flesh) in a blender until you have a smooth purée.
Sieve the mixture, discarding the bits of basil left behind in the sieve, making sure to press the pulp to extract all the flavours and juices.
Transfer to an ice-cream machine and churn according to the manufacturer’s instructions. Store in an airtight container.