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There is no short answer to this question. Do note that your cat will not become sick if you give it ice-cream. Your kitty will actually love to have some. Your cat can eat both normal ice-creams loaded with sugar or sugar-free ones having artificial sweeteners.
Your cat has no sweet tooth
If some ice-creams are lethal to dogs for containing xylitol, a compound toxic to dogs. Cats, however, does not get affected by this chemical. You will be surprised to know that cats have no sweet tooth as they cannot taste the sweet flavor. It means that even as your kitty licks ice-cream, the cat does not get the sweet taste. Your kitty will not respond to sweets if they see it. You must put it in front of them to elicit any reaction.
Cats get attracted to ice-cream not for its sweet taste, but for its milk. Your furball loves full milk but avoids the product's fat-free or skim versions. This is why cats ignore frozen yogurts. In short, the felines get attracted to the high carbohydrate and fat content of ice-cream. They love to drink full-fat milk but avoids the thinner ones. Although many foods have higher fat content and more carbohydrates compared to ice-cream, cats would go for the latter as they are familiar with the consistency and taste of milk. It also helps that full cream milk is a must for nutrition.
Ice-cream in small quantities are harmless to cats. You should not give ice-cream in large quantities to your kitty. The best portion is a tablespoon. If you give your cat a daily dose of ice-cream, then it may suffer from diarrhea.
Cats, like their human counterparts, could be lactose intolerant. It means they lack sufficient quantities of lactase enzyme to digest the milk lactose. Some cats, however, have more quantities of the enzyme compared to other cats. Kittens can consume the milk generated by their mothers. They produce substantial quantities of lactase enzyme when young. When the kittens grow up, more enzymes are required to digest proteins.
As milk is not included in the daily diet of your cat, there will be a reduction in lactase production. It means the milk will not be properly digested. This happens as its body realizes there is no requirement to digest or consume milk. If your cat does drink milk, it will come out undigested at the other end and cause a stink in the home. The lactose in the milk comes out along with the other waste. The milk may curdle if it comes in contact with your cat's stomach's digestive fluids. This may cause abdominal discomfort to your cat and gas formation.