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We derive a space-charge limit for a circularly symmetric bunched electron beam that has its center-of-mass slightly offset from the axis of a circularly symmetric coaxial conducting structure. The limit is a necessary condition for beam stability resulting from the following two forces acting on the center-of-mass of the electron bunch: (1) the radial deflecting force due to the induced surface charges and currents on the conducting structure and (2) the confining force due to the presence of an external solenoidal magnetic-focusing field. We show explicit calculations of the space-charge limits for beam-charge distributions, which in the radial direction have zero thickness, i.e., thin annular beams, and in the axial direction have either zero longitudinal thickness or Gaussian like distributions. Finally, we show how this limit is applicable to a previous design study of the University of California, Davis/Stanford Linear Accelerator Center 2.8-GHz coaxial ubitron oscillator.