Have ready an ice cream maker ( I use an electric Krups model, resembling this Cuisinart model with a 1 quart canister that requires pre-freezing for at least 24 hours). Arrange a 5 x 9 inch loaf dish (because it’s easier to scoop out of the loaf pan than the Krups canister) in the freezer making sure it is level and secure.
Put the sugar and water into a small saucepan. Cook, stirring, over medium heat until sugar is dissolved. Set aside to cool completely (you can refrigerate this syrup up to 4 days before continuing).
Combine the syrup, celery, lime juice, and lime zest in the blender. Purée the mixture (you may need to pulse to get it going at first) until the celery is finely ground. Transfer the mixture to a container, cover, and refrigerate overnight.
The next morning, strain the mixture through a fine sieve into another bowl. Leave it in the fridge for several hours to extract all the juice. Make sure the juice is well chilled before proceeding.
Pour the chilled mixture into the ice cream maker and process according to manufacturer’s instructions. Serve immediately (it will be soft) or scrape into the chilled dish, cover and freeze until firm or up to 2 days.