I know the existence of a celebrity, I know her name, but I don't met her personally and not a friend of her. I should use say 'Je la sais/connais'?

Bonjour Joan !

You would say - Je la ‘connais’.

The reason is the rule-

1. Connaître - To be used if we need to emphasise on “knowing a person, place, or a noun( used as an object)

2. Savoir - To be used if we emphasise on knowing how an action is being done/ completed ( with the help of a verb)

For example -> 

Knowing how to make/ prepare a dish uses “savoir” whereas Knowing the instructions/ recipe of the dish uses “ connaître”.

Hope this helps !

Hi Joan,

You definitely wouldn't say "Je la sais". I think you probably could say "Je la connais", but if you are afraid that might be misunderstood then you would have to be more specific - e.g. "Je sais qui c'est" = "I  know who she is".

You might find this article useful:

https://www.frenchtoday.com/blog/french-vocabulary/savoir-versus-connaitre-the-verb-to-know-in-french/

connaître means to cognize, so you should say "Je la connais"