I can see there's been a lot of excellent debate on this matter. I would add, just to help, that the verb saber comes usually with another verb o with a phrase. For example:
Yo sé leer. - I know how to read.
In this first case, saber is related to an ability.
Yo sé que hoy lloverá. - I know that it'll rain today.
In this other case, saber is related to facts that you know. But, in both cases, saber is related to a dominated subject.
Conocer is, on the other hand, a verb that should always be followed by an object (direct or indirect):
Yo conozco París. - I know Paris.
Yo te conozco. I know you.
And, as the rest of commenters have stated, conocer is sometimes a verb that denotes a general state of awareness or general knowledge. But, as we can see, when you KNOW (conocer) someone, or when you know Paris, it means that you genuinely know the person, or place, in a very profound way. This, evidently, will depend on the context. But I'd say that, in most cases, saber and conocer are only different in the way that they build their sentences - saber with phrases or verbs, conocer with objects.