Bebras tasks are considered to develop Computational Thinking (CT) and are currently used for this purpose in many studies. However, the relationship between Bebras tasks and CT is recent and, given the scarcity of validated instruments for assessing CT that are not associated with a specific learning environment, the expected development of CT through Bebras tasks has not been sufficiently evaluated, especially in Primary School. For this reason, we designed the ABC-Thinking program for the development of CT in which a set of Bebras tasks were selected and categorised according to the CT skills they were related to, specific lesson plans were designed to integrate these tasks in a 12-week curriculum, and teachers were trained to apply the program. Using the competent Computational Thinking test (cCTt) in a pre-post-test experimental design, we looked to determine whether Primary School students developed specific CT skills after the program, and, therefore, whether Bebras could be considered suitable for the development of this competence. The results show a significant development in specific CT skills, interesting findings in the gender differences in this development, and between high and low performers. Finally, qualitative data indicate the suitability of the ABC-Thinking program with respect to teachers' practice.