The recent discovery of quantized flux appeared to explain the extremelylong lifetime .of persistent currents in superconductors. The existence of the flux quantum indicated that the energy of a superconductor in a current carrying state is a local minimum with respect to current variations if the flux is an integer number of flux quanta. The stability is thus explained as a greatly reduced transition probability for the large current changes needed to lower the energy. Bloch and Rorschach have investigated the energetic stability of persistent currents in a long hollow superconducting cylinder, using the charged Bose gas model proposed by Schafroth. The criterion for stability was that the energy of the system could not be lowered by single particle transitions. They showed that the maximum stable magnetic field which could be trapped inside the cylinder was a function of the externally applied field and the sample dimensions, and could be less than the critical field Hc. In contrast to these results one would predict from Maxwell's equations, together with the infinite conductivity of a superconductor, that it should be possible to trap and maintain any field inside the cylinder as long as both the internal and external fields are smaller than the critical field. Measurements have been made of the maximum stable field inside a hollow tin cylinder (wall thickness d 1.5 mm, inner radius r = 7.5 mm, length L =12.5 mm) as a function of the externally applied field. The results seem to be in qualitative agreement with the Bose gas theory, in that the maximum internal field is a function of the external field. However, the theory predicts that when the external field is zero the maximum internal field should be d/r Hc = 0.2 Hc, while the experimentally determined value is 0.6 Hc. It is possible that these results are due to end effects, as the field at the cylinder walls is greater at the ends than at the center. The rate of decay of the internal field when the external field is changed indicates that these end effects may be important.