The magnetic flux density in the cavity of a sintered tube of Yttrium Barium Copper Oxide (YBCO) is measured as a function of H∥, the axial magnetic field impressed and removed after zero field cooling (ZFC). We note that, in a short thick-walled hollow cylinder of a weak-linked polycrystalline high-Tc superconductor, the magnetic field in the cavity of the tube, Hzhole, rises above the applied field H∥ when it is initially increasing. Consequently the standard approach to determine the critical current density from an ascending sweep of H∥ alone, or a descending sweep alone cannot be exploited. However the field dependence of the intergranular critical current density Jcm can be reliably obtained from the horizontal traversals of Hzhole vs H∥. Comparison of the experimental data with model calculations for the horizontal traversals of the hysteresis curves show that the intergranular critical current density in the YBCO tube has a Kim-like field dependence. The magnetization of the grains has a significant influence on the flux density in the cavity of the tube.