Excitons in organic semiconductors possessing a large oscillator strength demonstrate strong coupling with cavity modes at room temperature. A large Stokes shift in some organic semiconductors enriches and complicates the picture of the emission in strongly coupled systems of organic excitons and light. Here, we demonstrate strong coupling of excitons in 4,4-bis(N-carbazolyl)-1,1-biphenyl (CBP) and Tamm plasmons in the ultraviolet (UV) band, accompanied by a bright emission from the structure. Reflection measurements demonstrate the pronounced formation of the lower and upper polariton modes with Rabi splitting of the magnitude of 0.3 eV, and the emission peak experiences a substantial red shift with respect to the lower polariton mode. Both radiative and nonradiative decay rates in the Tamm plasmon CBP structure are increased with respect to a bare CBP. Such peculiar behavior is attributed to the simultaneous manifestation of strong coupling and weak coupling of the CBP molecule emitters to the Tamm plasmons.