We report experimental results for the Kapitza resistance RK between gold and superfluid He4, which were obtained by using very-high-resolution thermometry. The data imply that RK is singular at the superfluid transition temperature T<. Comparison with theory suggests that the singularity results from a hydrodynamic effect proposed by Landau, and that it is associated with the vanishing of the superfluid and normal-fluid currents at the boundaries. Recently a quantitative prediction of the singular part of this Kapitza resistance has been made based on dynamic renormalization-group theory. This theory requires no adjustable parameters and it agrees well with our data. The measurements of RK are independent of the heat flux Q only for reduced temperatures t=1-T/T< greater than a characteristic value tc. We find that tc is approximately proportional to Q, and that for t<tc the maximum value of (<RK/<Q)t is approximately proportional to 1/t. The Q-dependent contribution RKQ to RK can be represented by a function of the single variable Q/t. For large Q/t, this function saturates at a value near 0.35 cm2 K/W. We know of no theory that predicts the dependence of RK on Q.