From the dictionary blog:
When you want to colloquially express that you don’t care at all about something you might say “I couldn’t care less.” This phrase first popped up in British English at the turn of the 20th century and is still popular today. In the 1960s, a controversial American variant of this phase entered popular usage: “I could care less.” Many native English speakers, both in the UK and US, find this expression to be logically flawed. If you couldn’t care less, then you care so little about something that it would be impossible for you to care any less than you do. If you could care less, however, you are saying, literally, that it is possible for you to care less than you care now. Those who take issue with this believe this later variant says very little about your level of caring, and so eschew it.
Etymologists suggest that “I could care less” emerged as a sarcastic variant employing Yiddish humor. They point to the different intonations used in saying “I couldn’t care less” versus “I could care less.” The latter mirrors the intonation of the sarcastic Yiddish-English phrase “I should be so lucky!” where the verb is stressed.
The argument of logic falls apart when you consider the fact that both these phrases are idioms. In English, along with other languages, idioms are not required to follow logic, and to point out the lack of logic in one idiom and not all idioms is…illogical. Take the expression “head over heels,” which makes far less sense than the expression “heels over head” when you think about the physics of a somersault. It turns out “heels over head” entered English around 1400, over 250 years before “head over heels,” however, the “logical” version of this idiom has not been in popular usage since the late Victorian era.
The usage of “couldn’t care less” versus “could care less” is a very polarizing issue as you can see in British comedian David Mitchell’s rant, though both phrases are in popular usage. Because most modern English dictionaries define words and phrases using a descriptive approach, you’ll find both “couldn’t care less” and “could care less” in Dictionary.com. The lexicographers at Dictionary.com aim to record language as it is actually used, without judgment. That said, not everyone you encounter will be a lexicographer, so be aware that those in the camp of David Mitchell will cringe if you use “I could care less” in conversation.
Long story short..... both are correct.
It seems we truly are two cultures, separated by a common language. 
