Bring the water and sugar to a boil in a small pot until the sugar has dissolved. Allow the syrup to cool. Finely grate the green part of 2 of the limes and squeeze the juice from 3 and add this to the syrup. Add the coconut milk and stir.
Once this has cooled, process this in a small food processor with the avocado until you have a very smooth and soft mixture. It should be the thickness of whipped cream and almost a spreadable consistency.
If making by hand:
Pour the whipped mixture into a Ziploc bag. Place the bag on a tray and flatten it out and seal it. Freeze the bag for a few hours or overnight.
Once frozen, remove from the bag and break into chunks into the bowl of a food processor. Whiz the processor for 30 seconds or until smooth. You may need to let it soften a little at room temperature before processing it through the machine.
To keep it solid, scrape down the bowl of the food processor with a spatula and decant it into a metal container. Smooth the surface and freeze until solid and you are ready to serve. Otherwise, decant this into a piping bag with a large nozzle (I used a 16mm nozzle) and pipe it into your serving dish or cone.
If making with a machine:
Pour the mixture into the ice cream machine and churn until it is a scoopable consistency. Remove and freeze until ready to serve(to make a soft serve, serve it while it is still soft).
Notes
make this ice cream in advance to allow to freeze and refreeze.