The experimental results reported in this paper concern a very slow time-periodic axisymmetric oscillation of the entire Taylor vortex system with an azimuthal wave number m = 0 appearing as a secondary or higher time-dependent instability only. Therefore it is qualitatively different from other nonstationary states such as the wavy Taylor vortex flow or modulated wavy vortex flow. It is shown that this flow state is caused by the coupling strength of the phases of the underlying wavy flow modes. The observed oscillation frequencies range from 2 X 10(-2) to 3 X 10(-5) times the inner cylinder's angular velocity. Thus, we call this flow the very low frequency (VLF) mode. Up to now there has been no satisfactory theory which can explain the appearance of this mode.