Uniaxial bianisotropic medium is a generalization of the well-studied bi-isotropic and chiral media. It is obtained, for example, when microscopic helices with parallel axes are positioned in a host dielectric in random locations. Plane wave propagation in such a medium is studied and a simple solution for the dispersion equation and for the eigenwaves are found. As a numerical example, polarization properties of a transverse wave propagating in a uniaxial bianisotropic medium is considered. The results give a simple possibility to construct a polarization transformer with a transversely uniaxial chiral medium for changing the polarization of a propagating plane wave.