CRISPR-Cas is a form of adaptive sequence-specific immunity in microbes. This system offers unique opportunities for the study of coevolution between bacteria and their viral pathogens, bacteriophages. A full understanding of the coevolutionary dynamics of CRISPR-Cas requires knowing the magnitude of the cost of resisting infection. Here, using the gram-positive bacterium Streptococcus thermophilus and its associated virulent phage 2972, a well-established model system harbouring at least two type II functional CRISPR-Cas systems, we obtained different fitness measures based on growth assays in isolation or in pairwise competition. We measured the fitness cost associated with different components of this adaptive immune system: the cost of Cas protein expression, the constitutive cost of increasing immune memory through additional spacers, and the conditional costs of immunity during phage exposure. We found that Cas protein expression is particularly costly, as Cas-deficient mutants achieved higher competitive abilities than the wild-type strain with functional Cas proteins. Increasing immune memory by acquiring up to four phage-derived spacers was not associated with fitness costs. In addition, the activation of the CRISPR-Cas system during phage exposure induces significant but small fitness costs. Together these results suggest that the costs of the CRISPR-Cas system arise mainly due to the maintenance of the defence system. We discuss the implications of these results for the evolution of CRISPR-Cas-mediated immunity.