Yes. If an acid is very weak, it won't dissociate very well into it's product. [H+] [A-]. Which means that the conjugate base [A-] (since it doesn't like being in this form due to instability) will be more likely to steal an H+ and form the weak acid. So in sum...
Weak Acid --> Strong Conjugate Base.
Strong Acid --> Weak Conjugate Base.