Bring the water and sugar to the boil. Whisk on a low heat until the crystals have melted, then boil down for 5 minutes. When the syrup has cooled to a gentle heat, add the basil leaves, cover, and allow to cool.
Scrub the lime under the tap, dry it and zest it. Squeeze out the juice.
Peel the kiwis and cut them into pieces. Put them in the bowl of an electric mixer with the syrup (previously filtered), lime juice and vanilla essence. Mix until you have a smooth, even purée.
Add the lime zest. Mix again, then leave to set in a sorbet maker or in the freezer (stirring frequently to prevent crystals from forming).
Store the sorbet in the freezer in an airtight container.
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