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, and stir in the cream until smooth. Strain the mixture through a fine sieve set over a clean bowl. Cover and chill the mixture in the refrigerator, stirring occasionally, until cool, about 1 hour.
Pour the chilled mixture into an ice cream maker and freeze according to manufacturer's directions until it reaches "soft-serve" consistency. Transfer ice cream to a lidded plastic container then cover the surface with plastic wrap and seal. For best results, ice cream should ripen in the freezer for at least 2 hours or overnight.