We present the results of GALEX observations of 17 cool core (CC) clusters of galaxies. We show that GALEX is easily capable of detecting star formation in brightest cluster galaxies (BCGs) out to $z\ge 0.45$ and 50-100 kpc. In most of the CC clusters studied, we find significant UV luminosity excesses and colors that strongly suggest recent and/or current star formation. The BCGs are found to have blue UV colors in the center that become increasingly redder with radius, indicating that the UV signature of star formation is most easily detected in the central regions. Our findings show good agreement between UV star formation rates and estimates based on H$\alpha$ observations. IR observations coupled with our data indicate moderate-to-high dust attenuation. Comparisons between our UV results and the X-ray properties of our sample suggest clear correlations between UV excess, cluster entropy, and central cooling time, confirming that the star formation is directly and incontrovertibly related to the cooling gas.