Corynebacterium diphtheriae, the causative agent of diphtheria, produces the diphtheria toxin only when it is infected by phage \( \beta \). The phage carries the gene for the diphtheria toxin, and the toxin is responsible for the symptoms of diphtheria, such as throat inflammation and difficulty breathing.
Phage \( \beta \) integrates into the bacterium's genome and expresses the toxin gene. Without the phage, C. diphtheriae does not produce the toxin, even though the bacteria may still be present.
Thus, the correct answer is \( \boxed{(A)} \).