The most idiomatic way to say it in English while avoiding the troublesome "but" would be
Just because I'm stupid that doesn't mean I don't know.
In fact I think the whole expression should be taken as
だからって…ない with the negation being crucial (though sometimes implied*), and corresponds exactly to the English expression
"Just because … doesn't mean … ". It expresses that "Your assertion that A is a sufficient reason for B is not valid".
If you look at
いって (saying), you can understand it as "saying it is so because". The
ない then negates that assertion of causality: "Your saying that B is so because of A is not valid.", or "Just because of A (which is what you say) is not a valid reason for B.", which is equivalent to the English expression.
As sawa writes above, even in Japanese the meaning of って as "you say" is somewhat extraneous. I think it more easily becomes extraneous when expressed in English.
So your original sentence says that your being stupid is not a sufficient reason for (one's assumption of) your not knowing [some fact].
A asserts: You are stupid, therefore you don't know that fact.
B: That's not so. While I am indeed stupid, I do know that fact.
Therefore my being stupid does not imply that I do not know the fact.
Just because I am stupid does not mean I do not know the fact.
- In dainichi's example お客だからって態度が悪すぎる I find the negative in the すぎる where "excessive" implies "not appropriate"; "his attitude is too shitty" is the same as "his attitude should not be so shitty". Though I may be reaching here.