  We calculate the thermal Casimir effect for a dielectric tube of radius $R$
and thickness delta formed from a membrane in water. The method uses a
field-theoretic approach in the grand canonical ensemble. The leading
contribution to the Casimir free energy behaves as -k_BTL kappa_C/R giving rise
to an attractive force which tends to contract the tube. We find that kappa_C ~
0.3 for the case of typical lipid membrane t-tubules. We conclude that except
in the case of a very soft membrane this force is insufficient to stabilize
such tubes against the bending stress which tends to increase the radius.
