A numerical evaluation of the average superfluid density of 4He in a cylindrical pore near T a predicted by the Ginzburg-Pitaevskii-Mamaladze theory has been carried out as a function of pore radius and temperature. The results are presented in graphical form and in terms of an accurately fitting closed algebraic form.The only detailed prediction for the behavior of the superfluid density ps of 4He in a cylindrical pore of which we know is provided by the free-energyexpansion theory of Ginzburg and Pitaevskii 1 as modified by Mamaladze.2'$ Although frequent reference has been made in the literature to the temperature To at which ps disappears in the pore and its increasing depression below the bulk lambda temperature Ta as the pore radius is decreased, we are not aware of the publication of any calculation of the explicit dependence of as on T below To for comparison with experiment. § To fill this gap we have calculated the average value 0,p of p, over the cross-sectional area of the pore as a function of temperature and pore radius predicted by the Ginzburg-Pitaevskii-Mamaladze (GPM) theory.In the GPM theory p~(r) is given locally by malO(r)l 2, where m4 is the mass of the 4He atom, and ~O(r) is an order parameter which satisfies the differential equation -(h2/2m4) V24, -ag, +/31¢]2~ = 0(1) subject to the boundary condition that ~ should vanish at the walls of the