Whisk together the eggs, egg yolks, sugar, lime juice and lemon zest in a saucepan over medium heat until well-blended. Continuously stir the egg mixture with a wooden spoon until thickened, 7 to 8 minutes. The mixture should be thick enough to coat the back of the spoon. Remove from heat; stir in the single cream until smooth. Pass the mixture through a fine sieve set over a clean bowl. Cover and chill the mixture in the fridge, stirring occasionally, until cool, about 1 hour.
Pour the chilled mixture into an ice cream maker and freeze according to manufacturer's directions. Transfer ice cream to a lidded plastic container; cover surface with cling film and seal. For best results, ice cream should ripen in the freezer for at least 2 hours or overnight.