This paper provides a real example of applying COCOMO II as an estimation technique for the required software development effort in a safety-critical software application project following the DO-178C processes. The main goal and contribution of the case study is to support the research on software effort estimation and to provide software practitioners with useful data based on a real project. We applied the method as it is, by correlating the effort multiplier factors with the complexity and objectives introduced by the DO-178C level A application, resulting in an estimated effort. The rationales for each scale factor and effort multiplier selection were also described in detail. By comparing the estimated values with the actual required data, we found a magnitude of relative error (MRE) of 40% and provided alternatives for future work in order to increase the effort estimation accuracy in safety-critical software projects.