A new method for obtaining an orthogonal system of eigenwaves of an open cylindrical waveguide filled with a gyrotropic medium and located in free space is presented. The advantage of the method is that it enables one to explicitly represent the fields of eigenwaves, which correspond to the discrete and continuous parts of the eigenvalue spectrum of such a guiding structure. Orthogonality relations for the eigenwaves and the procedure of expanding an electromagnetic field in terms of these modal solutions are discussed. The limiting transition from the case of a closed cylindrical waveguide with a perfectly conducting wall and a coaxial cylindrical gyrotropic core to the case of an open waveguide is considered. To illustrate the completeness of the obtained system of eigenwaves, a given field is expanded in terms of the found discrete- and continuous-spectrum waves and then resynthesized by evaluating the corresponding expansion numerically. Perfect coincidence between the initially specified field and the result yielded by this evaluation is demonstrated.