Blend the sugar and the basil leaves in a small food processor until coarse.
In a large bowl, using an electric whisk, cream the mascarpone with the yoghurt, then add the basil-sugar. Whisk until combined.
Either: Pour it into an ice cream machine and follow the instructions of the manufacturer, then freeze.
Or: If you don't have an ice cream maker, transfer the mixture into a large, shallow plastic container. Cover and freeze for 2 hours, then beat the cream with a hand- held blender (in the container) until smooth. Return to the freezer. Repeat this process 2-3 times to ensure that the ice crystals properly break down and the ice cream becomes smooth.
10 minutes before serving, transfer the container to the refrigerator - it will be easier to spoon the ice-cream out.